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AN INTRODUCTION TO THE

THEORY

OF STATISTICS

OTHER BOOKS ON STATISTICS, &c

A statistical pnmer

The advanced theory of statistics

Ranh correlation methods

Exercises in theoretical statistics

The design and analysis of experiment

Sampling methods for censuses and surveys

Statistical method in biological assay

The mathematical theory of epidemics

Biomathematics

Prohahihty and the weighing of evidence

F. N. DAVID
M G. KENDALL
M G KENDALL
M. G. KENDALL
M. H QUENOUILLE
F. YATES
D. J. FINNEY
N. T. J. BAILEY
C. A B SMITH
I. J GOOD

Griffin's Statistical Monographs and Courses:

No. 1 The analysis of multiple time-senes m h. quenouille
No. 2* A course in multivariate analysis M. G. KENDALL

Particulars from the Publishers

AN INTRODUCTION TO THE

THEORY OF STATISTICS

The Late G. UDNY YULE, M.A., F.R.S.

Formerly Reader tn StatisHcs, University of Cambridge

and

M. G. KENDALL, M.A., Sc.D.

Professor of Statistics, University of London

FOURTEENTH EDITION
REVISED AND ENLARGED

Third Impression

CHARLES GRIFFIN & COMPANY LIMITED
LONDON

Copynght ^ All editions
CHARLES GRIFFIN & CO LTD
42 DRURY LANE, LONDON, WC2
'Ml nghts reserved

First edition

1911

Second edition

1912

Third edition

1916

Fourth edition

1917

Fifth edition

1919

Sixth edition

1922

Seventh edition

1924

Eighth edition

1927

Ninth edition

1929

Tenth edition 1932

{Two impressions)
Eleventh edition . 1937

Twelfth edition 1939

T hirteenth edition 1 944

(Five impressions)
Fourteenth edition 1950

Second

impression 1953

Third

impression 1958

/ 6.9886

PRINTED IN GREAT BRITAIN BV CHARLES BIRCHALL & SONS, LTD., 17 , JAMES STREET, LIVERPOOL, 2

PREFACE TO THE FOURTEENTH EDITION

The first edition of this book, by Mr. Udny Yule, was based on the
courses given during his tenure of the Newmarch lectureship in Statistics
at University College, London. It appeared in 1911 and ran to ten
editions by 1935, at which stage Mr. Yule felt that a complete revision
was necessary and asked me to undertake it. The eleventh edition,
under our joint names, appeared in 1937. Two further editions and
several reprints have subsequently been necessary, and translations have
appeared in Portuguese and Spanish.

This fourteenth edition is again a substantial revision. Although
fewer than fifteen years have passed since the last revision, so much has
happened in the statistical world in the meantime that Mr. Yule and I both
felt that the usefulness of the book would be increased by some further
changes. Most of the alterations are additions, but the treatment of the
theory of attributes, which in earlier editions occupied five chapters,
has been condensed into three to make room for the new material.

The major additions fall into two groups. Chapters 21-23 expand
the former treatment of small-sample theory and give an introduction to
the practical problems of sampling. Chapters 25-27 give an account
of index-numbers and the elementary theory of time-series. Chapter 13
on practical problems of correlation has also been re-written. Additions
have been made in the remaining chapters to keep the treatment abreast
of new discoveries, some of the examples have been modernised and some
further exercises added. The list of references has been omitted because
a much more extensive bibliography has now appeared in volume 2 of
my Advanced Theory of Statistics,

Mr. Yule's original object was to make this book a systematic intro-
ductory course on statistical methods suited to those who possess only
a limited knowledge of mathematics. I have never lost sight of this object.
The amendments in this edition are not due to any alteration in our design ;
they are necessitated by the development of our subject. In particular,
the book now aims at covering the theoretical part of the syllabus laid
down by the Royal Statistical Society for its Certificate. Although I
assume responsibility for the new material, the general plan of the

THEORY OF STATISTICS

revision was agreed between Mr Yule and myself and once again I have
been able to draw on his experience and advice. A bald acknowledgment
of this kind completely fails to express the extent of my indebtedness
to him.

The tables of “ Student’s ” t are reproduced by permission of the late
W S Cosset and the proprietors of Metron ; those of the F- and
distributions by permission of Professor R. A. Fisher and Messrs. Oliver
and Boyd to whom my grateful thanks are due I shall be indebted to
any reader who calls my attention to errors or obscurities.

M.G.K.

London,

March, 1950

NOTE TO SECOND IMPRESSION

George Udny Yule died on June 26th, 1951 in his eighty-first year.
Among his many and varied contributions to the advancement of statistics
this book has a high place. I shall try to carry it forward in the spirit
in which he originally conceived it.

This printing is the same as the fourteenth edition except for the correc-
tion of a few minor misprints.

M.G.K.

London,

March, 1953

NOTE TO THIRD IMPRESSION

As the study ot statistics develops — and it has grown very substantially
over the past few years — a book such as this has to contend with numerous
competitors. I am gratified to note that it appears to be maintaining its
place, not only in Great Britain but in many other countries of the
world. This is, in my opinion, chiefly due to the basic soundness of Udny
Yule’s original approach to the subject. At all events, new' generations of
students continue to use the book, and another large impression of this
edition has been demanded long before I expected it. As m the previous
impression, only minor amendments have been required to remove a few
misprints and ambiguities and to bring some of the references up to date.

M.G.K.

London,

November, 1957

CONTENTS

Chapter

Page

NOTES ON NOTATION AND ON TABLES FOE FACILITATING
STATISTICAL WORK

INTRODUCTION ....

VyJ'^HEORY OF ATTRIBUTES : BASIC IDEAS

\Jir ^ ASSOCIATION OF ATTRIBUTES

3 MANIFOLD CLASSIFIC.\TION .

Frequency-distributions .

AVERAGES AND OTHER MEASURES OF LOCATION

MEASURES OF DISPERSION ....

MOMENTS AND MEASURES OF SKEWNESS AND KURTOSIS

•^REE IMPORTANT THEORETICAL DISTRIBUTIONS : THE

BINOMIAL, THE NORMAL, AND THE POISSON

9 CORRELATION AND REGRESSION

10 NORMAL CORRELATION

11 FURTHER THEORY OF CORRELATION

12 PARTIAL CORRELATION.

CORRELATION AND REGRESSION : SOME PRACTICAL PROBLEMS

14 MISCELLANEOUS THEOREMS INVOLVING THE CORRELATION
COEFFICIENT

15 Sp«PfE CURVE FITTING

iCT PfiELIMINARY NOTIONS ON SAMPLING ....
iff THE SAMPLING OF ATTRIBUTES : LARGE SAMPLES .

IS, idlE SAMPLING OF VARIABLES : LARGE SAMPLES .

THE SAMPLING OF VARIABLES : LARGE SAMPLES CONTINUED

^ THE DISTRIBUTION

^ THE SAMPLING OF VARIABLES : SMALL SAMPLES

xiii

102

125

151

169

199

237

253

281

310

326

340

366

386

413

437

459

482

Vll

viii

THEORY OF STATISTICS

Chaptg^ Page

PIE ANALYSIS OF VARIANCE 503

33* SOME PROBLEMS OF PRACTICAL SAMPLING .... 530

INTERPOLATION AND GRADUATION 555

25' INDEX NUMBERS 590

26. TIME-SERIES. . 610

27 TIME-SERIES (2) . . . 638

APPENDIX TABLES 663

(1) Normal curve. (2) areas under the normal curve. (3) Significance
points of W stable. (5) Significance points of the variance-
ratio F. (6) Significance points of the distribution of z.

ANSWERS TO THE EXERCISES 675

INDEX 691

NOTES ON NOTATION AND ON TABLES FOR
FACILITATING STATISTICAL WORK

A. Notation

The reader is assumed to be familiar with the commoner mathematical
signs, e,g. those for addition and multiplication. We shall also employ
the following symbols, all of which are in general use —

The factorial sign

The symbol n 1 , read factorial n” means the number
Ix2x3x ... X («~2) x(n— l)xn

Factorial n is by some writers expressed by the symbol but this
notation appears to be falHng out of use in favour of n I, probably owing
to the greater ease with which the latter form can be printed and type-
written.

The combinatorial sign

The symbol means the number of ways in which r things can be
chosen from n things, e.g., ^20^3 is the number of ways in which a hand
of cards can be dealt from an ordinary pack of 52 cards.

In most textbooks on algebra it is shown that

A more modern symbol is

n 1

and we shall use this form occasionally.

The summation sign

The sum of n numbers , , , Xn is written S [xt), read sum Xf

t=X

from one to n,'' i.e.

r—n

Where no ambiguity is likely to arise, the suffix r and the limits
written above and below S are omitted, e.g. the above sum would be
written simply l^ix), it being understood from the context that the
summation extends over the n values.

Many writers use the Roman letter S instead of X.

THEORY OF STATISTICS

The Greek alphabet

As the letters of the Greek alphabet will often be used as symbols, we
give for convenience the names of those letters.

Small

letter

Capital

letter

Name

Small

letter

Capital

letter

Name

alpha

beta

gamma

omicron

delta

€

epsilon

rho

zeta

a, s

sigma

eta

tau

theta

upsilon

iota

<!>

phi

kappa

chi {pron. ki)

lambda

psi

omega

B. Calculating Tables

For heavy arithmetical work a calculating machine is invaluable ;
but owing to their cost machines are, as a rule, beyond the reach of the
student.

For a great deal of simple work, especially work not intended for
publication, the student will find a slide rule exceedingly useful : par-
ticulars and prices will be found in any instrument-maker's catalogue.
For greater exactness in multipl 5 dng or dividing, logarithms are almost
essential.

The student will derive invaluable aid from Barlow’s Tables of Squares,
Cubes, Square-roots, Cube-roots, and Reciprocals of all Integral Numbers
up to 10,000 (E. & F. N. Spon, London and New York), which are useful
over a wide range of statistical work.

C. Special Tables of Functions useful in Statistical Work

The tables at the end of this book will cover most of the student's
ordinary requirements. The more advanced student will find it useful
to have Tables for Statisticians and Biometricians (Cambridge University
Press) — particularly Part I. Research workers vdll wish to have Fisher
and Yates* Statistical Tables for Biological, Agricultural and Medical
Research (Oliver and Boyd).

D, References to the Text

Each section in the book is distinguished by a number in heavy type
consisting of the number of the chapter in which the section occurs
prefixed to the number of the section in that chapter and separated from
it by a period ; e.g., 7.13 means the thirteenth section of Chapter 7, and
10.1 refers to the first section of Chapter 10. The Introduction, which

THEORY OF STATISTICS

precedes Chapter 1, is for this purpose regarded as Chapter 0, e.g.. 0.26
refers to the twenty-sixth section of the Introduction. References to
sections are given simply by the number of the sections, e.g., " We saw
in 8.3 means “ We saw in the third section of Chapter 8/’

Similarly, equations, tables, examples, exercises, diagrams and references
are distinguished first of all with the number of the chapter in which they
occur and then, separated by a period, with their serial number within
the chapter, e.g,, ‘‘ Table 6.7 refers to the seventh table in Chapter 6,
and “ Equation (17.8) refers to the eighth equation of Chapter 17.
These figures are in ordinary type

This simple notation saves a good deal of unnecessary wording. To
facilitate quickness of reference we sometimes give pages as well.

A distinction is drawn between examples, which are given in the text
for purposes of illustration, and exercises, which are set at the end of the
chapter for the student to work out for himself.

INTRODUCTION

Number and measurement

0.1 Western civilisation is pervaded by ideas of number' and measure-
ment. Even the events of our everyday life are inextricably bound up
with them. We have only to picture a race which cannot count or measure
trying to run the Bank of England or control the milk market, or even
understand the sporting columns of the daily press, to realise how deeply
rooted numbers are in the complex activities of the modern world.

0,2 Science itself is particularly indebted to numerical expression.
As organised knowledge has increased, the necessity for precision has
become greater, and in the formulation of precise statements number and
measurement have played a leading part. The desire for quantitative
expression was first felt in the physical sciences, but it has now spread into
nearly all branches of knowledge. The movement is by no means com-
plete, however, and may be seen at work to-day. As a significant instance
we may note that courageous attempts are being made to subject the
process of thought itself — that last stronghold of the contentious and the
mysterious — to quantitative inquiry.

0.3 Many people, in fact, have been led by their enthusiasm for
numerical data to regard knowledge of a non-quantitative kind as hardly
deserving the name “ knowledge ’’ at all. Towards the close of the nine-
teenth century it was possible for Lord Kelvin to say : When you can
measure what you are speaking about and express it in numbers you know
something about it ; but when you cannot measure it, when you cannot
express it in numbers, your knowledge is of a meagre and unsatisfactory
kind.'" This remark has often been quoted with an approval which it does
not altogether deserve — it does not, for example, do justice to the work of
Darwin and Pasteur, to name only two of Kelvin's contemporaries. But
there can be no den 5 dng that it expresses a point of view which many
people will endorse.

Numerical data

0.4 The desire for precision, in fact, leads investigators of all kinds,
from the atomic physicist to the business man, to express the facts about
that part of the universe which interests them in a quantitative way.
Numerical data have come into being not only in the laboratory and the
study, but in the counting-house, the sales department, the Board Room
and the legislative assembly. It is difficult to see how our society could be

xiii

XIV

THEORY OF STATISTICS

organised without them. Where the Jews and the Romans were content
with occasional censuses for military or fiscal purposes^ the progressive
modern state finds itself under the necessity of keeping a close and quanti-
tative eye on all that goes on within or without its frontier. A country
which does not do so may be fairly regarded as backward. In a typical
phrase, Anatole France summed up this point of view when he said of the
Chinese’ ''Tant quhls ne se seront pas compt^s, ils ne compteront pas” —
if they don’t count they won’t count.

Statistics concerned with numerical data

0.5 There are certain features of numerical data, no matter in what
branch of knowledge they originate, which may call for a special type of
scientific method to treat them and elucidate them. This is known as
" Statistical method,” or more briefly, as “ Statistics.” It does not,
however, embrace the study of numerical data of every kind, and before
we attempt a formal definition of its nature and scope, it is necessary to
give some words of explanation.

Effects and causes

0.6 One of the principal aims of Science is to trace, amidst the tangled
complex of the external world, the operation of what are called ” laws ” —
to interpret a multiplicity of natural phenomena in terms of a few funda-
mental principles. A knowledge of the operation of these laws enables us
to talk of “ cause ” and ” effect.” The metaphysical problems associated
with these words need not detain us, but since in the sequel we shall often
use them, it is proper to explain that we adopt them as a convenient way
of expressing serviceable and familiar ideas. We shall be dealing with
the everyday world, where ” law ” and ” cause ” have significant and
important connotations.

0.7 With this convention, we may say that any physical event, and
in particular that described by quantitative data, is produced by the
operation of one or more causes. The number of causes which produce any
particular effect may be, and usually is, extremely large. For instance,
the height of a man is causally Jinked with his race, his ancestry, his
habitation, his diet during youth, his age, his occupation, and at any given
moment even with his position and the time of day.

0.8 Experiment, the great weapon of scientific inquiry, derives its power
from the ability of the experimenter to replace such complex systems of
causation by simple systems in which only one causal circumstance is

^ David {II Samuel, 24) numbered the people of Israel and called down a plague by
doing so. He counted 800,000 valiant men who drew the sword, and though the text
IS not entirely clear it seems likely that Divine disapproval was directed against the
militaristic purpose of the census, not the census itself. We are told later that 70,000
men died of the resulting pestilence, so it looks as if there was no ban on counting dead
men.

INTRODUCTION

allowed to vary at a time. This is perhaps an ideal, but it is one which
is closely approached with the technique of modern laboratory practice.

0.9 Let us, howevei, turn for a moment to social science, as the parent
oi the methods termed '' statistical,'' and consider its characteristics as
compared, say, with physics ot chemistry. One characteristic stands out
so markedly that attention has been repeatedly directed to it by
** statistical " writers as the source of the peculiar difficulties of their
science — the observer of social facts cannot experiment, but must deal with
circumstances as they occur, apart from his control. The simplification open
to the experimenter being impossible, the observer has, in general, to deal
with highly complicated cases of multiple causation — cases in which a
given result may be due to any one of a number of alternative causes or
to a number of different causes acting conjointly.

0.10 A little consideration will show that this is also characteristic of
observations in other fields. The meteorologist, for example, is in almost
precisely the same position as the student of social science. He can
experiment on minor points, but the records of the barometer, thermo-
meter and rain gauge have to be treated as they stand. With the biologist,
matters are somewhat better. He can and does apply experimental
methods to a very large extent, but frequently cannot approximate closely
to the experimental ideal ; the internal circumstances of animals and plants
too easily evade complete control. Hence a large field (notably the study
of variation and heredity) is left in which methods of experiment have to
be supplemented by other methods. The physicist and chemist, finally,
stand at the other extremity of the scale. Theirs are the sciences in which
experiment has been brought to its greatest perfection. But even so, there
is still scope for the application of statistical treatment in these sciences.
The methods available for eliminating the effect of disturbing circumstances,
though continually improved, arq not, and cannot be, absolutely perfect.
The observer himself, as well as the observing instrument, is a source of
error ; the effects of changes of temperature, or of moisture, or pressure,
and draughts, vibration, etc., cannot be completely eliminated.

0.11 It is with data affected by numerous causes that Statistics is mainly
concerned. Experiment seeks to disentangle a complex of causes by
removing all but one of them, or rather by concentrating on the study
of one and reducing the others, as far as circumstances permit, to a com-
paratively small residuum. Statistics, denied this resource, must accept
for analysis data subject to the influence of a host of causes, and must
try to discover from the data themselves which causes are the important
ones and how much of the observed effect is due to the operation of each.

Definitions

0.12 In the light of the foregoing discussion we may accordingly give
the following definitions —

xvi

THEORY OF STATISTICS

By Statistics we mean quantitative data affected to a marked extent
by a multiplicity of causes.

By Statistical Methods we mean methods specially adapted to the
elucidation of quantitative data affected by a multiplicity of causes.

By Theory of Statistics or, more briefly. Statistics we mean the
exposition of statistical methods.

(It will be observed that the same word may be used both for the
science and for the raw material on which it works. This dual use
gives rise to no confusion in practice, but the distinction is worth bearing
in mind.)

Use of statistic

0*13 This is perhaps the appropriate place, to remark that there has
recently come into use the singular form statistic."' This is the name
given to a particular kind of estimate compiled from observations, iisually
according to some algebraical formula. In this book we shall not meet
the term until we reach the theory of sampling (Chapter 18) and shall
there use it in a restricted sense.

History of the word statistics ”

0.14 In their present meaning the words “ statistics," statistician"
and " statistical " are barely a century old. They have, however, been
in use longer than that, and it is instructive to consider the process by
which they have reached their present meaning.

0*15 The words "statist," "statistics," "statistical," appear to be
all derived, more or less indirectly, from the Latin status, in the sense,
acquired in mediaeval Latin, of a political State,

0*16 The first term is, however, of much earlier date than the two others.
The word " statist " is found, for instance, in Hamlet (1602)i, Cymbeline
(1610 or 1611),* and in Paradise Regained (1671).3 The earliest occurrence
of the word " statistics " yet noted is in The Elements of Universal
Erudition, by Baron J. F. von Bielfeld, translated by W. Hooper) M.D.
(3 vols., London, 1770), One of its chapters is entitled Statistics, and
contains a definition of the subject as " The science that teaches us what is
the political arrangement of all the modern states of the known world," *
" Statistics " occurs again with a rather wider definition in the preface to
A Political Survey of the Present State of Europe by E. A, W. Ziramermann,®

» Act 5, SC.2. » Act 2, sc. 4. a 4^

^ We cite from Br W. F. Willcox, Quarterly Puhhcations of the American Statistical
Association, vol. 14, 1914, p, 287,

* ^immermann's work appears to have been written in English, though he was a
German and Professor of Natural Philosophy at Brunswick.

INTRODUCTION

xvii

issued in 1787. It is about forty years ago/' says Zimmermann, that
that branch of political knowledge, which has for its object the actual and
relative power of the several modem states, the power arising from their
natural advantages, the industry and civilisation of their inhabitants, and
the wisdom of their governments, has been formed, chiefly by German
writers, into a separate science. ... By the more convenient form it has
now received . . . this science, distinguished by the new-coined name of
statistics, is become a favourite study in Germany (p. ii) and the
adjective is also given (p. v) : To the several articles contained in this
work, some respectable statistical writers have added a view of the
principal epochas of the history of each country."'

0.17 Within the next few years the words were adopted by several
writers, notably by Sir John Sinclair, the editor and organiser of the first
Statistical Account of Scotland,^ to whom, indeed, their introduction has
been frequently ascribed. In the circular letter to the Clergy of the Church
of Scotland, issued in May 1790,^ he states that in Germany " ' Statistical
Inquiries," as they are called, have been carried to a very great extent,"
and adds an explanatory footnote to the phrase " Statistical Inquiries " —
''or inquiries respecting the population, the political circumstances, the pro-
ductions of a country, and other matters of state."" In the " History of the
Origin and Progress of the work, he tells us, " Many people were at first
surprised at my using the new words. Statistics and Statistical, as it was
supposed that some term in our own language might have expressed the
same meaning. But in the course of a very extensive tour, through the
northern parts of Europe, which I happened to take in 1786, 1 found that in
Germany they were engaged in a species of political inquiry, to which they
had given the name of Statistics ... as I thought that a new word might
attract more public attention, I resolved on adopting it, and I hope that it
is now completely naturalised and incorporated with our language." This
hope was certainly justified, but the meaning of the word underwent rapid
development during the half-century or so following its introduction.

0.18 " Statistics "" (statistik), as the term was used by German writers
of the eighteenth century, by Zimmermann and by Sir John Sinclair,
meant simply the exposition of the noteworthy characteristics of a state,
the mode of exposition being — almost inevitably at that time — ^pre-
ponderantly verbal. The conciseness and definite character of numerical

1 Twenty-one vols., 1791-99.

* Statistical Account, vol. 20, Appendix to “ The History of the Origin and Progree#
...” given at the end of the volume.

* Loc, ctt., p. xiii.

* The Abriss der Sfaatswissenschaft der Europdischen Eeicke (1749) of Gottfried
Achenwall, Professor of Politics at Gottingen, is the volume in which the word
” statistik ” appears to be first employed, but the adjective ” statisticus ” occurs at a
somewhat earlier date in works wntten in Latin.

XVlll

THEORY OF STATISTICS

data were recognised at a comparatively early period — more particularly
by English writers — but trustworthy figures were scarce. After the
commencement of the nineteenth century, however, the growth of official
data was continuous, and numerical statements, accordingly, began more
and more to displace the verbal descriptions of earlier days. '' Statistics
thus insensibly acquired a narrower signification, viz. the exposition of
the characteristics of a State by numerical methods. It is difficult to
ssiy at what epoch the word came definitely to bear this quantitative
meaning, but the transition appears to have been only half accomplished
even after the foundation of the Royal Statistical Society in 1834. The
articles in the first volume of the Journal, issued in 1838-39, are for the
most part of a numerical character, but the official definition has no
reference to method. Statistics,"' we read, may be said, in the words
of the prospectus of this Society, to be the ascertaining and bringing
together of those facts which are calculated to illustrate the condition
and prospects of society.” It is, however, admitted that ” the statist
commonly prefers to employ figures and tabular exhibitions.”

0.19 Once the first change of meaning was accomplished, further
changes followed. From the name of a science, the word was transferred
to those series of figures on which it operated, so that one spoke of vital
statistics, shipping statistics, and so on. It was then applied to the
similar numerical data which occurred in other sciences, such as anthro-
pology and meteorology. By the end of the nineteenth century we find
” statistics of mental characteristics in man,” ” statistics of children
under the headings bright-average-dull,” and even ” an examination of
the characteristics of the Virgilian hexameter with statistics.” The
development of the meaning of the adjective ” statistical ” and the noun
” statistician ” was naturally similar.

0.20 Perhaps the most abstract use of the word occurs in the theory
of thermodynamics, wherein one speaks of entropy as proportional to the
logarithm of the statistical probrjbiliiy of the universe — a definition which
no statesman would be unwilling to admit to lie completely outside his
pur\iew. But it is unnecessary to multiply instances to show that the
word statistics ” is now entirely divorced from ” matters of State.”

The theory of statistics

0.21 The theory of statistics as a distinct branch of scientific method
is of comparatively recent growth. Its roots may be traced in the work
of Laplace and Gauss on the theory of errors of observation, but the
study itself did not begin to flourish until the last quarter of the nineteenth
century. Under the influence of Galt on and Karl Pearson remarkable
progress was made, and the foundations of the subject were laid in the
next tliirty years — as it has turned out, very securely. The subject has

INTRODUCTION

XIX

not, however, yet reached a stage whereat a cut-and-dried exposition of
its methods can be given. Research, particularly into the mathematical
theory of statistics, is rapidly proceeding, and fresh discoveries are being
made with a rapidity which makes it difficult to keep pace with them.
It may, however, help the student to appreciate the work of later chapters
if we sketch in brief general terms the field of statistical theory as it now
exists.

The collection of data

0.22 The first question which the statistician has to consider is the
collection and assembling of his data. In many fields, such as economics
and sociology, he cannot prepare the data himself but has to get what
he can from such sources as official statistics, which are usually prepared
with an object differing from his own. Such information is therefore
rarely all that one could wish. Investigator A, studying the sugar
market, finds that the official figures run cane and beet sugar together.
Investigator B, wanting to compare prices over a period of years, finds
that during the war period 1939-1945 there is a gap in the information.
Investigator C, wishing to study poverty, has to content himself with
indirect figures such as those of wage levels and unemployment. But
however incomplete the data may be, and however tangentially pertinent
to his inquiry, the investigator must take what he can get and be thankful

0.23 In other cases, and particularly in meteorology, biology and
psychology, he can produce his own data or borrow those of other investi-
gators similarly engaged. He does not merely take his figures from some
source or other ; he is instrumental in their production, and within limits
can control their nature so as to bring them to bear directly on his inquiry.

It might be thought that the only qualities required for such work are
an ability to count or measure and a reasonable care But this is not so
Once outside the laboratory the investigator is beset with a swarm of
practical difficulties. We might illustrate the point by referring to the
troubles of an investigator who wished to find out how many dairy cows
there were in a certain parish. He took the simplest course and went to
all the farms in the parish and asked the occupier how many cows he had.
Farmer A said that he had fifteen, but had sold eight and was waiting
for the buyer to come and fetch them. Farmer B had about twenty.”
Farmer C obviously could not be bothered and said the first figure which
came into his head ; and so on. It is clear that the result of such an
inquiry would be to give a quite illusory figure. One of the duties of the
practising statistician is to design his inquiries so as to minimise this kind
of error.

0.24 A full discussion of such matters lies outside the scope of this
book, but we have given them more than a passing mention in order to
introduce one very necessary caution.

THEORY OF STATISTICS

The reliability of data must always be examined before any attempt
is made to base conclusions on them. This is true of all data, but
particularly so of numerical data, which do not carry their quality written
large upon them. It is a waste of time to apply the refined theoretical
methods of statistics to data which are suspect from the beginning.

The treatment of data

0.25 Having obtained his data and satisfied himself that they are
reliable enough to permit him to proceed, the statistician must then “ lick
them into shape."' He must decide on some form of arrangement and
presentation, reduce them to a convenient scale of units, and so on ; in
short, he must work on his raw material until it is ready for the application
of his prepared tools

0.26 The only process of treatment to which attention need be called
is that of condensation. The mind is incapable of grasping the significance
of a large mass of figures. If, therefore, the quantity of data available
is of any size, some process of condensation is necessary to enable the
mind to appreciate the picture which the data represent.

Suppose, for instance, we are discussing the stature of a thousand men,
and have as data the height of each man to the nearest inch. Our raw
material then consists of a thousand sets of figures ranging from four feet
to seven feet, or thereabouts. Only the supermind could look over these
figures and grasp their essentials. Nor would the position be met by
rearranging the figures in order of magnitude. To get a clear picture of
the situation some condensation is necessary, and in this case it can be
carried out easily by grouping together all the men whose heights lie in a
certain range, say of three inches. Our total range of three feet is then
replaced by twelve sub-ranges, each of three inches, and we may
summarise the data by giving the numbers of men who fall into the twelve
sub-ranges. In short, we have replaced our original thousand figures by
twelve.

0.27 It will be clear that in so doing we have sacrificed a certain
amount of information. Twelve figures cannot possibly tell us as much as
a thousand. It may very well be, however, that the information in the
twelve is all that we require ; the lost information may be irrelevant to
the inquiry. Such a case would happen if we wanted to know, to an inch
or so, what was the height exhibited by the greatest number of men.

0.28 The process of condensation thus sacrifices information but gives
us instead a very necessary clarity and adaptability for manipulation.
How far the process is carried in any particular case will depend on how far
the disadvantages of the sacrifice are offset by the advantages of the
darity.

INTRODUCTION

XXI

Summarising and descriptive statistics

0.29 The process of summarising which we have just described may
be carried a great deal further, and leads to a branch of theory which has
very important practical appUcations.

The reader is probably familiar already with the idea of an “ average
value” and with its use in compressing into a single number the results of
a series of observations. Such quantities are, in fact, the result of sum-
marising to the greatest possible extent ; they are summaries in which the
statistician has distilled the information of a diffuse mass of figures into a
single drop, so to speak.

0.30 There is a wide demand for such summarising numbers, and a
good deal of this book will be devoted to considering them from one aspect
or another. They give a convenient bird's-eye view of what is sometimes
a complex and confusing whole. Special sciences have evolved special
quantities of this type to meet their own needs. For instance, the econo-
mist has invented various kinds of index numbers to express in a short-
hand way complicated changes in prices; and the psychologist has devised
coefficients to express the reactions of an individual mind to a sequence of
tests.

0.31 The remarks we made in 0.27 and 0.28 apply here with additional
force. It must never be forgotten that in summarising we omit. Part of
the statistician's task is to see that we do not omit too much.

0.32 The problem of describing a complicated set of data in as few
terms as possible is facilitated by the use of mathematical functions.
Suppose, for instance, that in the thousand men of 0.26 we assumed that
the number of men (y) of height x inches varied as the square of x —
frankly a most improbable result, but one which will serve for the purposes
of illustration. Then we may describe the data completely by an equation
of the form —

y=^ax'^

where a is a constant to be determined from the data. Knowing a we can
find the number of men of any given height.

0.33 In this case it rather looks as if we have condensed all the
information into a single number a without losing any of it. But that is
not so. What we have done is to replace the set of a thousand figures by
an assumption about their nature. We have lost none of the information
because we assumed, in using the equation, that the information was of
a type known to us already.

0.34 It is found in practice that many sets of data may be very con-
veniently expressed by mathematical functions. The question as to which

xxn

THEORY OF STATISTICS

functions are the most suitable for purposes of description leads to some
interesting theory, some of which will be dealt with later and some of which
is of an advanced character lying outside the scope of an Introduction to
the Theory of Statistics, Such functions are particularly helpful in the
theory of sampling.

Analysis of data

0.35 When the statistician has arranged and compressed his data into
a suitable form, or decided on the functions and evaluated the quantities
which he has chosen to describe them, the first stage of his inquiry is
finished. It may be that he would wish to take it no further ; for instance,
if he is preparing an index number for the economist he may wish to hand
over the number to that person without comment, for him to make such
use of it as he thinks fit. More frequently, however, he has prepared the
data for his own use as a statistician. He then proceeds to the next
stage, that of analysis and elucidation of the causal system which gave rise
to them.

0.36 The methods for such purposes are very numerous. In this
brief review we need only point out the importance of the investigation of
relationship, the theory of which bulks very large in statistical literature.
If two events are related there is usually, though not always, some causal
nexus between them. The problems of the investigation of relationship
between phenomena lead to the theory of dependence, contingency and
correlation, and the formulation of various coefficients to measure the
extent to which one set of events depends upon another.

Sampling

0.37 When we wish to discuss the properties of an aggregate we may
be prevented by practical or theoretical reasons from examining every
single member of it. For example, in considering the stature of the male
inhabitants of the United Kingdom we cannot measure every man,
because of the time and trouble involved ; and in considering the scores
of a roulette wheel we cannot examine every score, because the number
is practically infinite and observations can be continued as long as the
wheel lasts.

0.38 We do not despair, nevertheless, of being able to gain some
knowledge of the aggregate. Where we cannot take the whole we do the
best we can and try to obtain a selection of members. This selection is
called a sample.

0.39 It is clear that a sample will not tell us everything about the
parent aggregate from which it is derived. Nevertheless, most people have
a feeling, and we shaE see later in this book that under certain conditions
the feeling is a justifiable one that the sample will give us some information

INTRODUCTION

XXlll

about the parent. Values calculated from the sample may be taken to be
estimates of values in the parent, to a degree of approximation which
becomes closer as the sample gets larger; and even where the sample is
small we can sometimes draw inferences of a general nature about the
parent.

0.40 We are rarely, if ever, able to reason from the sample to the parent
with the categorical certainty of a mathematical proof. Our inferences
will usually be expressed in terms of probabilities. Moreover, we shall find
it much easier to reject a hypothesis than to accept it. Our inferences
will generally be not of the type the hypothesis H is true/' or even
*'the hypothesis H is probably true/' but of the type hypotheses A,
B and C are probably untrue, but we see no reason to doubt hypothesis

For example, suppose we take a sample of a thousand men from the
population of the United Kingdom and find their average height to be
5 ft 8 in. What can we say about the average height of the population as
a whole ^ We cannot give it with any certainty. We cannot even say,
with certainty, that it lies within, say, one inch of 5 ft 8 in. What we can
say, assuming that the sampling technique is sound, will be something to
the effect that a hypothesis which supposes that the mean of the whole
population is greater than 5 ft 9 in. or less than 5 ft 7 in. is probably
incorrect, but that the data are consistent with the supposition that the
mean lies between those limits.

0.41 The theory of sampling is thus closely bound up with the theory
of probability. The many problems which arise in this connection are
among the most interesting and at times the most difficult which science
and philosophy can offer. It is only fair to warn the student that there
still exists an important difference of opinion among scientific men about
the validity of certain types of statistical inference. In this book we have,
so far as we could, avoided these contentious matters, but the advanced
student will have to be prepared to face them sooner or later.

The popular attitude towards statistics

0.42 Finally, to conclude this introduction we may, perhaps, refer to
the popular mistrust of statistics and statistical methods.

The layman's attitude towards statistics is admirably summed up in
the remark that mankind is divided into two parts, those who say that
figures can prove anything and those who assert that they can prove
nothing. It must be admitted that this attitude is not unreasonable.
From the advertisement hoarding, from the electioneering platform, from
the partisan press, and from a dozen other sources, the man in the street is
bombarded with tendentious figures put forward to support some ex park
statement. Sometimes such figures are justifiably used to form a basis for
the arguments which are built upon them ; more often they give a specious

3odv

THEORY OF STATISTICS

picture of the truth, which may be due to ignorance or inadvertence, but
has also been known to be occasioned by a deliberate wish to mislead.
The layman is well aware of this fact. His attitude in distrusting all
arguments based on figures is that of a reasonable man, who has not the
training to distinguish for himself the true from the false, and is therefore
inclined to suspect everything.

0.43 We are not concerned here with the vindication of statistics in
the public view We have alluded to the matter in order to remind the
student that statistical methods are most dangerous tools in the hands of
the inexpert. Few subjects have a wider application ; no subject requires
such care in that application. Statistics is one of those sciences whose
adepts must exercise the self-restraint of an artist.

CHAPTER ONE

THEORY OF ATTRIBUTES

BASIC IDEAS

Attributes and variables

1.1 The methods of statistics, as defined in the Introduction, deal with
quantitative data alone. The quantitative character may, however,
arise in two difierent ways.

In the first place, the observer may note only the presence or absence
of some attnbute in a series of objects or individuals and count how many
do or do not possess it. Thus, in a given population,^ (a useful general
term for the aggregate of objects under discussion, the extent and nature
of which should always be kept in mind) we may count, if we are dealing
with human beings,* the number of blind and seeing, or of Europeans and
non-Europeans ; if it is a population of coin-tosses, the number of heads
and tails ; if a population of pea-plants, the number of tails and dwarfs.
The quantitative character, in such cases, arises solely in the counting.

In the second place, the observer may note or measure the actual
magnitude of some variable character for each of the objects or individuals
observed. He may record, for instance, the ages of persons at death,
the prices of different samples of a commodity, the statures of men, the
numbers of petals in flowers. The observations in these cases are
quantitative ah initio.

1.2 The methods applicable to the former kind of observations, which
may be termed statistics of attributes ”, are also applicable to the
latter, or statistics of variables.” A record of statures of men, for
example,, may be treated by simply counting all measurements as tall that
exceed a certain limit, neglecting the magnitude of any excess, and
stating the numbers of tall and short (or more strictly not- tall) on the basis
of this classification. Similarly, the methods that are specially adapted to
the treatment of statistics of variables, making use of each value recorded,
are available to a greater extent than might at first sight seem possible for
dealing with statistics of attributes. For example, we may treat the
presence or absence of the attributes as corresponding to the changes of a
variable which can only possess two values, say 0 and 1. Or, we may
assume that we have really to do with a variable character which has been

^ In the present edition we have substituted this ’less technical and more usual
term for the logical term universe used in preceding editions.

THEORY OF STATISTICS

crudely classified, as suggested above, and we may be able, by auxiliary
hypotheses as to the nature of this variable, to draw further conclusions.
But the methods and principles developed for the case in which the observer
only notes the presence or absence of attributes are the simplest and most
fundamental, and are best considered first. This and the next two
chapters are accordingly devoted to the Theory of Attributes.

Classification with reference to attributes

1.3 The objects or individuals that possess the attribute, and those
that do not possess it, may be said to be members of two distinct classes,”
the observer classif 5 dng ” the population observed. In the simplest
case, where attention is paid to one attnbute alone, only two comple-
mentary classes are formed. If several attributes are noted, the process
of classification may, however, be continued indefinitely. Those that do
and do not possess the first attribute may be reclassified according as they
do or do not possess the second, the members of each of the sub-classes
so formed according as they do or do not possess the third, and so on,
every class being divided into two at each step. Thus the members
of the population of any district may be classified into males and females ;
the members of each sex into sane and insane ; the insane males, sane
males, insane females and sane females into blind and seeing. If we
were dealing with a number of peas {Fisum sativum) of different varieties,
they might be classified as taU or dwarf, with green seeds or yeUow seeds,
with wrinkled seeds or round seeds, so that we should have eight classes —
tall with round green seeds, tall with round yellow seeds, tall with wrinkled
green seeds, tall with wrinkled yellow seeds, and four similar classes of
dwarf plants.

1.4 It may be noticed that the fact of classification does not necessarily
imply the existence of either a natural or a clearly defined boundary
between the two classes. The boundary may be wholly arbitrary, e.g.,
where prices are classified as above or below some special value, barometer
readings-as above or below some particular height. The division may also
be vague and uncertain : sanity and insanity, sight and blindness, pass into
each other by such fine gradations that judgments may differ as to the
class in which a given individual should be entered. The possibility of
uncertainties of this kind should always be borne in mind in considering
statistics of attributes * whatever the nature of classification, however,
natural or artificial, definite or uncertain, the final judgment must be
decisive ; any one object or individual must be held either to possess the
given attribute or not.

Dichotomy

1.5 A classification of the simple kind considered, in which each class
is divided into two sub-classes and no more, has been termed by logicians
clmsification, or, to use the more strictly applicable term, division by

THEORY OF ATTRIBUTES

dichotomy (cutting in two). The classifications of most statistics are not
dichotomous, for most usually a class is divided into more than two sub-
classes, but dichotomy is the fundamental case. In Chapter 3 the relation
of dichotomy to more elaborate {manifold, instead of twofold or dichoto-
mous) processes of classification, and the methods applicable to some such
cases, are dealt with briefly.

1.6 For theoretical purposes it is necessary to have some simple notation
for the classes formed, and for the numbers of observations assigned
to each.

The capitals A, B, C, . . , will be used to denote the several attributes.
An object or individual possessing the attribute A will be termed simply
A. The class, all the members of which possess the attribute A, will
be termed the class A. It is convenient to use single symbols also to
denote the absence of the attributes A, B, C, , . We shall employ the
Greek letters a, y, . . . Thus if A represents the attribute blindness,
a represents sight, i.e., non-bhndness ; if B stands for deafness, /? stands
for hearing. Generally a” is equivalent to '' noi-A,'’ or an object or
individual not possessing the attribute A , the class a is equivalent to the
class none of the members of which possesses the attribute A,

1.7 Combinations of attributes will be represented by juxtapositions
of letters. Thus if, as above, A represents blindness, B deafness, AB
represents the combination blindness and deafness. If the presence and
absence of these attributes be noted, the four classes so formed, viz. AB
Ap, aB, afi, include respectively the blind and deaf, the blind but not deaf,
the deaf hut not blind, and the neither blind nor deaf. If a third attribute
be noted, e.g. insanity, denoted say by C, the class ABC includes those
who are at once deaf, blind and msane, A By those who are deaf and blind
but not insane, and so on.

Any letter or combination of letters like A, AB, aB, ABy, by means
of which we specify the characters of the members of a class, may be
termed a class symbol.

Class-frequencies

1.8 The number of observations assigned to any class is termed, for
brevity, the frequency ” of the class, or the class-frequency.*’ Class-
frequencies will be denoted by enclosing the corresponding class-symbols
in brackets. Thus, {A) denotes the number of A’s, i.e., objects possessing
attribute A ; (a/?C) denotes the number of a/?C’s, i.e. objects possessing
attribute C but neither A nor B : and so on for any number of attributes.

Order of classes and class-frequencies

1.9 The classes obtained by noting, say, n attributes fall into natural
groups according to the numbers of attributes used to specify the respective
classes, and these natural groups should be borne in mind in tabulating

THEORY OF STATISTICS

the class-frequencies. A class specified by r attributes may be spoken of
as a class of the rih order and its frequency as a frequency of the rth
order. Thus AB, AC, BC are classes of the second order; (^4), [A^),
{aBC)y [AByD], class-frequencies of the first, second, third and fourth
orders respectively.

1.10 Class frequencies should, in tabulating, be arranged so that
frequencies of the same order and frequencies belonging to the same
aggregate are kept together Thus the frequencies for the case of three
attributes should be grouped as given below, the whole number of observa-
tions denoted by the letter N being reckoned as a frequency of order zero,
since no attributes are specified.

Order 0

Order 1

(A)

(B)

(C)

(a)

(A)

(r)

Order 2

(AB)

(AC)

(BC)

(^A)

(Ay)

(By)

(aB)

(aC)

(AC)

(a/S)

(ay)

(Ay)

Order 3

(ABC)

[(xBC)

(ABy)

(xBy)

(AfiC)

(xjSC)

(A^y)

(xjffy)

The total number of class-frequencies

1.11 In such a complete table for the case of three attributes, twenty-
seven distinct frequencies are given : 1 of order zero, 6 of the first order,
12 of the second and 8 of the third.

In general, for n attributes, there are distinct class-frequencies, if we
count AT as a frequency of order 0. To demonstrate this, let us consider
the number of classes of different orders.

Of order 0 there is one class N,

Of order 1 there are 2n classes, for classes of this order contain only one
symbol, and each of the n attributes contributes two symbols, one of the
type A and one of the type a.

Of order 2 there are

x22 classes, for each class contains two

symbols, two attributes can be chosen from n m

n(n—l)

ways, and each

pair gives rise to 2^ different frequencies of the types (AB), (AjS), (ccB)
and (ayj).

Similarly, it may be seen that of order r there are

n(n-l) . . . (n-r+1) ^^^^
rl

classes.

THEORY OF ATTRIBUTES

Hence, the total number of class-frequencies is

. . .

2 rl

and this is the binomial expansion of (l+2)»»=3'».

It is clear that if n is at all large the number of class-frequencies will be
very great. For instance if n=6, the number is 729.

1.12 Fortunately, however, the class-frequencies are not independent
of one another, and it is not necessary, in order to specify the data com-
pletely, to give every class-frequency.

In the first place, let us note the simple result that any class-frequency
can always he expressed in terms of class^frequenctes of higher order. For
the whole number of observations must clearly be equal to the number of
A*s added to the number of a's, i.e,

N^{A) + {a) .... (1.2)

Similarly, the number of A's is equal to the number of ^I's which are
B's added to the number of A*s which are /?'s, i.e.

{A)=^{AB) + {AJ3) .... (1.3)

Similarly,

{AB)^{ABC)+(ABy) . . . (1.4)

and so on.

Ultimate class-frequencies

1.13 It follows at once from the result we have just given that every
class-frequency can be expressed in terms of the frequencies of the highest
order, i.e., of order n. For any frequency can be analysed into higher
frequencies, and the process need stop only when we have reached the
frequencies of the highest order. For example, with three attributes,

. {A)={AB)+{Afi)

={ABC)+{A By) + (A^C) + (A/iy)

The classes specified by n attributes, i.e. those of the highest order, are
termed the ultimate class-frequencies.

Our result may then be expressed in the form . Every class-frequency
can be expressed as the sum of certain of the ultimate class-frequencies. To
specify the data completely it is, therefore, only necessary to give the
ultimate class-frequencies.

Example 1.1 — (See F. Warner and others, “ Report on the Scientific
Study of the Mental and Physical Conditions of Childhood,'' Parkes
Museum, 1895.) A number of school-children were examined for the
presence or absence of certain defects of which three chief descriptions
w^ere noted : A, development defects ; B, nerve signs ; C, low nutrition.

THEORY OF STATISTICS

Given the following ultimate frequencies, find the frequencies of the
classes defined by the presence of the defects, i.e. those involving the
Roman letters A, B, C but not the Greek letters a, 7 , including the
whole number of observations N —

(ABC)

(ocBC)

(A By)

281

(aBy)

670

(A^C)

(afiC)

(APy)

453

(a^y)

8310

The whole number of observations N is equal to the grand total :
iV=10,000.

The frequency of any first-order class, e.g. (A), is given by the total of
the four third-order frequencies the class-symbols for which contain the
same letter —

{A BC) + {A By) + {A^C)+{A^y) ={A) =877

Similarly, the frequency of any second-order class, e.g. (AB), is given
by the total of the two third-order frequencies the class-symbols for which
both contain the same pair of letters —

{ABC) + {ABy)=^ {A B) =338

The complete results are —

10,000

(AB)

338

(A)

877

(AC)

143

(B)

1,086

(BC)

135

(C)

286

(ABC)

The number of ultimate class-frequencies

1.14 The class-frequencies of highest order each contain n symbols.
Now each letter corresponding to a particular attribute may be written
in two ways • A or a, B or etc. Hence the total number of possible
symbols is

2x2x2x2x2x2x2x . . . =2«

and this is the number of ultimate class-frequencies.

Hence the 3^ frequencies may all be expressed in terms of the 2 «
ultimate frequencies. For example, if ^== 6 , the 729 frequencies can be
written in terms of 64 ultimate class-frequencies, which specify the data
completely.

The ultimate frequencies are, however, not the only set which specify
the whole of the data. In fact, any set will serve the purpose provided
that {a) they are 2*^ in number, and ( 6 ) they are algebraically independent ;
that is to say, when they are written symbolically no one can be expressed
in terms of some or all of the others.

We may call such a set of frequencies a fundamental set.

THEORY OF ATTRIBUTES

Positive attributes

1.15 The attributes denoted by capitals ABC , . may be termed

positive attributes, and their contraries, denoted by Greek letters, negative
attributes. If a class-symbol includes only capital letters, the class may
be termed a positive class ; if only Greek letters, a negative class. Thus
the classes A, AB, ABC are positive classes; the classes a, atf, ccfiy,
negative classes.

If we make a certain dichotomy with regard to a definite attribute A —
such as male sex, blindness or blue eyes — ^it may be of practical importance
to note a possible distinction in the nature of the class not-J. The
complementary class may, in fact, either be equally definite — ^female sex,
abihty to see — or it may be a mere heterogeneous remainder, as in our
last instance — not-blue-eyed, the not-blue-eyed being brown-eyed, grey-
eyed, or even possessing no eyes at aU.

Logically, this distinction is difficult to maintain, but practically it is
of some importance. The statistical data in official returns are almost
always classified according to positive and clearly defined attributes.
For example, we are given the numbers of persons d 3 dng from typhoid,
not the numbers who did not die of typhoid ; the number of acres under
grass, not the number of acres not under grass.

1.16 The positive class-frequencies form a fundamental set in the sense
of 1.14 ; that is to say, they specify the data completely. They are
algebraically independent ; no one positive class-frequency can be
expressed whoUy in terms of the others. Their number is, moreover, 2^
as' may be readily seen from the fact that if the Greek letters are struck
out of the symbols for the ultimate classes, they become the symbols for
the positive classes, with the exception of afiy , . . for which N must be
substituted.

Example 1.2. — Given the positive class- frequencies of Example LI, to
find all the class-frequencies.

The data are —

iV=10,000; (^)=877; (B)=1086; (C)=286; (.4S)=338;

(^C)=143 ; (5C)=135 ; (^SC)-57.

We have —

{AB)^{ABy)+{ABC)

338-(.IBy)+57

i.e.

(4Sy)=281

Similarly, from [AC) and [BC) we find —

[A^C)^m

(aJ5C)=78

THEORY OF STATISTICS

This gives us the three ultimate class-frequencies which contain only
one Greek letter. For the others,

{a^C)={pC)-{ApC)

=286-135 -86
=65

Similarly, we have —

{APy)=A53

(aFy)=670

Finally,

[aPy)=={Py)-{Apy)

Hy)-{By)-{Apy)
==f^-{C)-\{B)-[BC)]~{APy)
=10,000-286-951 -453
=8310

We can now calculate any class-frequency by expressing it in terms of
the ultimate class-frequencies, e.g.

(a7)=(aBr)H-(a/?y)

=670-1-8310

=8980

1.17 The data encountered in practice are rarely dichotomised according
to more than three or four variables, and the student should experience
little dijQ&culty in expressing any class-frequency in terms of the known
class-frequencies, either directly, or by first finding the ultimate class-
frequencies and then expressing the desired frequency in terms of them.

It is, however, interesting to note the general result that the class
symbols can be treated as operators and multiplied together like algebraical
quantities. Let us write A,N for the operation of dichotomising N
according to A, and write

A.N^{A)

which is the symbolic way of saying that if we dichotomise N according to
A we get a class-frequency equal to {A). We can similarly put

a . N={a)

Adding these two, and putting A. N+cc.N equal to (.4 +a) . iV, we have —

{A+a).N^N

so that we may take

In any symbolic expression we can therefore replace the operators ^ or a
by r— a, l—i4, respectively.

Furthermore, since {AB)—A . (S)=5 . {A), we may take the symbol

THEORY OF ATTRIBUTES

J5 . iV to be the dichotomy of N according to both A and B, and equate
it to {AB). A little reflection will show that the operative symbols
therefore obey the ordinary laws of algebra and in particular may be
multiplied together.

For example, we have —

(ayff)=a^ . N^(l-A){l . N

^{l-A-B+AB) ,N

==N^{A)^(B)+iAB) , . . . (L5)

And, similarly,

{(xfiy) =a/?7 . N

=={l-^A){l^B){l^C),N
={1-A-B^C+AB+BC+AC-ABC) . N
^N^{A)^(B)-^{C)+{AB)+{AC)+{BC)^{ABC) . . (1.6)

Similar results could, of course, be obtained by step-by-step sub-
stitution ; for instance,

^N^{A)^{B) + {AB)

Consistence

1.18 Any class-frequencies which have been or might have been observed
within one and the same population may be said to be consistent with
one another. They conform with one another, and do not in any way
conflict.

The conditions of consistence are some of them simple, but others are
by no means of an intuitive character. Suppose, for instance, the following
data are given —

1000

(AB)

{A)

525

(AC)

147

(B)

312

(BC)

470

(ABC)

— there is nothing obviously wrong with the figures. Yet they are
certainly inconsistent. They might have been observed at different
times, in difierent places or on different material, but they cannot have
been observed in one and the same population. They imply, in fact, a
negative value for {<xpy ) —

[oLpy] =1000 -525 -312-470 +42 + 147 +86 -25
=1000-1307+275-25
= -57

Clearly no class-frequency can be negative. If the figures, conse-
quently, are alleged to be the result of an actual inquiry in a definite
population, there must have been some miscount or misprint.

THEORY OF STATISTICS

Condition for consistence

1.19 It IS, m fact, the necessary and sufficient condition for the con-
sistence of a set of independent class-frequencies that no ultimate class-
frequency be negative. It is necessary for the obvious reason that no
class-frequency occurring by counting real attributes can be negative ;
it is sufficient because, given any non-negative set of 2”' numbers, we can
always imagine a real population with n dichotomies which should have
these numbers for its ultimate class-frequencies, and it is impossible for
this real population to give inconsistent results.

Hence to test the consistence of a set of 2” algebraically independent
class-frequencies we need only calculate the ultimate class-frequencies and
ascertain whether any one is negative If it is, the data are inconsistent.
If no ultimate frequency is negative, the data are consistent.

1.20 For data given by a heterogeneous collection of class-frequencies,
consistence is best tested by actually calculating the ultimate frequencies.
We saw in 1.15, however, that the positive class-frequencies hold a peculiar
position in that many data encountered in practice are given entirely in
terms of them alone. It may be useful to consider the consistence
conditions for this type of material.

If two attributes are noted there are four ultimate frequencies {AB),
{Ap), (aB), {<xp). Expressing them in terms of positive classes we find
the following conditions —

{AB) ^0

{AB)^{A)^{B)^N
{AB) < (.4)

(AB) < (B)

(1.7)

The third and fourth merely express the fact that the number of members
which are both A and B must not be greater than the number of .4’s or
B’s separately. The second inequality is perhaps not so obvious.

1.21 For three attributes the conditions that the eight ultimate
frequencies are not negative will be found to lead to the following —

(ABC) > 0 \

(ABC)>(AB)+(AC)-(A) [

(ABC)^(AB)+(BC)-iB) \

(ABC) ^ (AC)+(BC)-(C) )

(ABC) < (AB)

(ABC) < (AC)

(ABC) ^(BC)

(ABC) < (^B)+(^C)+(BC)-(.4)-(B)-(C)+.¥

These are not of a new form. They can all be derived from inequalities
(1.7) by “ specifying the population” ; that is to say, by considering one

THEORY OF ATTRIBUTES

of the inequalities as holding in a sub-population. For instance, from the
condition {AB) < we have in the population of y's {A By) < {Ay)
which is equivalent to

{AB)-^{ABC) ^(A)-{AC)
or the second equahty of (1.8).

1,22 If we express the condition that the lower limits to {ABC) given
by (1.7) must be not greater than the upper limits given by (1.8) we
obtain 16 further inequalities. All but four of them are of the type
already found, but there are four new ones —

{AB)+{AC)+{BC)^{A)+{B)MQ-^ ]

{AB)+{AC)~{BC) ^{A)
(AB)-^{AC)+(BC) ^{B)
^{AB)+{AC)+{BC) ^{C) )

( 1 . 10 )

Incomplete data

1.23 We can now take up the question of the inferences which may be
drawn from data which, though giving us a certain amount of information
in the shape of class-frequencies, yet are insufficient to enable us to
calculate all the class-frequencies.

The form of the consistence conditions shows that a knowledge of
certain class-frequencies allows us to assign limits to others, even though
we may not be able to find the actual values of those others. The follow-
ing will serve as illustrations of the statistical uses of the conditions —
Example 1.3. — Given that (A)=(.B)=(C)=|iV and 80 per cent of the

C's, find the limits to the percentage

1 _o-8 -0-75

0-8+0-75 -1
1 -0-8 +0-75

1 +0-8 -0-75

limit greater than unity ; hence they
(c) we have —

— that is to say, not less than 55 per cent nor more than 95 per cent of
the 5*s can be C’s.

are B's, 75 per cent of are

of B*s that are C's.

The data are : =0 • 8

and the conditions (1.10) give —
{a) 2{BC)IN^

ib) >

{d) <

(a) gives a negative limit and (^^) a
may be disregarded. From (d) and

THEORY OF STATISTICS

Example 1.4— If a report gives the following frequencies as actually
observed, show that there must be a mispnnt or mistake of some sort, and
that possibly the misprint consists m the dropping of a 1 before the 85
given as the frequency (J5C) —

N 1000

(A)

510

{AB)

189

{B)

490

(AC)

140

(C)

427

(BC)

From (1.10) we have —

[BC) >510+490+427-1000~-189-140
>98

But 85 <c: 98, therefore it cannot be the correct value of {BC),

If we read 185 for 85 all the conditions are fulfilled.

Example 1.5. — In a certain set of 1000 observations (^)=45, (jB)= 23,
fC)=14. Show that whatever the percentages of B's that are ^'s and of
C's that are A*Sy it cannot be inferred that any B’s are C's.

The first two conditions of (1 . 10) give the lower limit of (BC) which is
required. We find —

N N N

The first limit is clearly negative. The second must also be negative,
since {AB) jN cannot exceed 0-023 nor (AC) jN, 0-014. Hence we cannot
conclude that there is any limit to (BC) greatei than 0. This result is
indeed immediately obvious when we consider that, even if all the B*s
were A's, and of the remaining 22 A's 14 were C's, there would still be
8 A*s that were neither B's nor C*s.

1.24 The student should note the result of the last example, as it
illustrates the sort of result at which one may often arrive by applying the
conditions (1.10) to practical statistics. For given values of N, (A), (B),
(C), (^IB) and {AC), it will often happen that any value of (BC) not
less than zero will satisfy the conditions (1.10), and hence no true
inference of a lower limit is possible. The argument of the type So
many .4*3 are B*s and so many B*s are C*s that we must expect some 4[*s
to be C's must be used with caution.

1-25 Where the data are not given in terms of the positive or of the
ultimate class-frequencies, and cannot readily be thrown into such a
form, the device illustrated in the following example is often useful —

THEORY OF ATTRIBUTES

Example 1.6. — Among the adult population of a certain town v50 per
cent of the population are male, 60 per cent are wage-eamers and 50
per cent are 45 years of age or over. 10 per cent of the males are not
wage-earners and 40 per cent of the males are under 45. Can we infer
anything about what percentage of the population of 45 or over are
wage-earners ?

Denoting the attributes male, wage-earner and 45 years old or more
by A, B and C, respectively, and letting N—lOO for convenience, we
have —

(.4) =50
(5) =60
(C)=50
(4^)= 5
(4y)=20

We require the limits, if any, of (-BC).

Let us note first of all that we are given 6 class-frequencies (including
N), If we knew two more, independent of these 6, the problem would
be completely determinate, for we should have 2® class-frequencies.

Let us therefore put

(a^y) =:x
(ABC)=y

We can then solve for the ultimate class-frequencies and get

(A By) =45 — y
{A^q =30 -y

{aBq=x -is

{Afiy) = y -25
(aBy) =30— x
((a/?C)=35— a;

The condition that these must be non-negative gives us conditions on x
and y. In fact, from {ocBC) and {ccBy) we get

15 <30

and from {ApC) and (A^y),

25 < y < 30

the conditions from the other frequencies being included in these limits
to X and y.

Now (BC)={4BC)-h{aBC)

=y-l-A;— 15

and hence, from the limits to x and y,

THEORY OF STATISTICS

Consequently, the percentage of the population 45 years old or more
(50 per cent of the total population) who are wage-earners lies between
50 and 90 per cent.

It is worth while examining whether these limits are the na^^o^^est
possible which can be assigned with the available data , and it is easy to
see that they are. For if x===l5 and y =25, (BC)=25 ; and if x=30 and
y=:30, (SC) =45. There is nothing in the conditions of the problem to
prevent x andy, and hence (SC), from reaching the limiting values, and
thus no narrowing of the limits is possible.

SUMMARY

1. A collection of individuals may be divided into two classes according
to whether they do or do not possess a particular attribute. This process
is called dichotomy.

2. Continued dichotomy according to n attributes gives rise to 3^
classes.

3. The frequencies in these classes can be expressed in terms of the 2«
ultimate class frequencies, or of the 2»» positive class frequencies.

4. Given 2» independent class-frequencies, all the class-frequencies may
be calculated by simple arithmetic^ processes.

5. The necessary and sufficient condition for the consistence of a set
of independent class-frequencies relating to a particular population is that
no ultimate class- frequency which may be calculated from them is
negative.

6. In view of the practical importance of the positive class-frequencies,
the form of the consistence conditions is expressed solely in terms of such
frequencies.

7. The conditions may be applied to the examination of inaccurate or
incomplete data. For the latter they may allow us to assign limits to
an unknown class-frequency.

EXERCISES

1.1 The following are the numbers of boys observed with certain classes
of defects amongst a number of school-children. A denotes development
defects ; nerve signs ; C, low nutrition.

{ABC)

149

(aBC)

204

{ABy)

738

(ocBy)

1,762

{A^C)

225

171

{A^y)

1.196

{apy)

21,842

Find the frequencies of the positive classes.

THEORY OF ATTRIBUTES

1,2 The following are the frequencies of the positive classes for the girls
in the same investigation —

23,713

{AB)

587

(A)

1,618

(AC)

428

(B)

2,015

(BC)

335

770

(ABC)

156

Find the frequencies of the ultimate classes.

1.3 (Figures from Census, England and Wales, 1891, voL 3) Convert
the census statement as below into a statement in terms of [a) the positive,
(b) the ultimate class-frequencies. ^ =bliiidness, B=deaf-mutism, C=
mental derangement.

29,002,525

(A By)

(A)

23,467

(Aj3C)

380

(B)

14,192

(aBC)

500

97,383

(ABC)

1.4 Show that if A occurs in a larger proportion of the cases where
B is than where B is not, then B will occur in a larger proportion of
the cases where A is than where A is not : i.e. given {AB) l{B)'>[Afi) J(^),
show that (AB) /{A) > (aB) /(a).

1.5 Given that

(^)=(a)=(J5)=(/J)=iiV

show that

{AB)=(afi). {Afi)={aB)

1.6 Given that

(^)=(a)={B)=(^)=(C) = (r)=i2V

and also that

{ABC)={afiy)

show that

2{ABC) ^{AB) +{AC)+(BC) - JiV

1.7 Measurements are made on a thousand husbands and a thousand
wives. If the measurements of the husbands exceed the measurements of
the wives in 800 cases for one measurement, in 700 cases for another,
and in 660 cases for both measurements, in how many cases will both
measurements on the wife exceed the measurements on the husband ?

1.8 100 children took three examinations. 40 passed the first, 39 passed
the second and 48 passed the third. 10 passed all three, 21 failed all three,
9 passed the first two and failed the third, 19 failed the first two and passed
the third. Find how many children passed at least two examinations.

Show that for the question asked certain of the given frequencies are
not necessary. Which are they ?

l6 THEORY OF STATISTICS

Show further that the data are not sufficient to permit of the deter-
mination of the ultimate class-frequencies.

1.9 (Lewis Carroll, A Tangled Tale, 1881) In a very hotly fought
battle 70 per cent at least of the combatants lost an eye, 75 per cent at
least lost an ear, 80 per cent at least lost an arm and 85 per cent at least
lost a leg. How many at least must have lost all four ?

1.10 Show that for n attributes A, B, C, . , . M,

{ABC . . . M) ^ {(^)+(S)+(C)+ . . . +(M)}-(^-~l)iV

where N is the total frequency ; and hence generahse the result of
Exercise 1.9.

1.11 In a free vote in the House of Commons, 600 members voted. 300

Government members representing English constituencies (including
Welsh) voted in favour of the motion. 25 Opposition members repre-
senting Scottish constituencies voted against the motion. The Govern-
ment majority among those who voted was 96. 135 of the members

voting represented Scottish constituencies. 18 Government members
voted against the motion. 102 Scottish members voted in favour of the
motion. The motion was carried by 310 votes. Analyse the voting
according to the nationality of the constituencies and party.

1.12 In a war between White and Red forces there are more Red soldiers
than White ; there are more armed Whites than unarmed Reds ; there
are fewer armed Reds with ammunition than unarmed Whites without
ammunition. Show that there are more armed Reds without ammunition
than unarmed Whites with ammunition

1.13. If, in an urban district 817 per thousand of the women between 20
and 25 years of age were returned as '' occupied ” at a census, and 263
per thousand as married or widowed, what is the lowest proportion per
thousand of the married or widowed that must have been occupied ?

1.14 If, in a series of houses actually invaded by smallpox, 70 per cent
of the inhabitants are attacked and 85 per cent have been vaccinated, what
is the lowest percentage of the vaccinated that must have been attacked ?

1.15 Given that 50 per cent of the inmates of an institution are men,
60 per cent are aged '' (over 60), 80 per cent non-able-bodied, 35 per
cent aged men, 45 per cent non-alile-bodied men, and 42 per cent non-
able-bodied and aged, find the greatest and least possible proportions of
non-able-bodied aged men.

1.16 The following are the proportions per 10,000 of boys observed for
certain classes of defects amongst a number of school-children,
development defects, B;=nerve signs, D “mental dullness.

THEORY OF ATTRIBUTES

N =10,000 (D)=789

{A)== 877 (^J5)=338

(B)= 1,086 {BD)==455

Show that some dull boys do not exhibit development defects, and state
how many at least do not do so.

1.17 The following are the corresponding figures for girls —

N =10,000 (Z))=689

(4)= 682 (^J5)=248

(B)= 850 (Bi))=363

Show that some defectively developed girls are not dull, and state how
many at least must be so.

1:18 Take the syllogism All A*s are B's, aJl B*s are Cs, therefore all
A's are express the premises in terms of the notation of the preceding
chapter, and deduce the conclusion by the use of the general conditions
of consistence.

1.19 Dodhe same for the syllogism All A*s are B’s, no B's are C’s,
therefore no A*s are Cs,**

1.20 Given that (^)=(B)=(C)=JiV. and that (AS)/iV=(AC) /i\r=/>,
find what must be the greatest and least values of p in order that we may
infer that (BC) IN exceeds any given value, say q,

1.21 Show that .if

( 2 = 3 ,

N N N

and

{AB)_{AC)_{BC)

N N N ^

the value of neither x nor y can exceed J.

1.22 A market investigator returns the following data. Of 1000 people
consulted, 811 liked chocolates, 752 liked toffee and 418 liked boiled
sweets ; 570 liked chocolates and toffee, 356 liked chocolates and boiled
sweets and 348 liked toffee and boiled sweets ; 297 liked all three. Show
that this information as it stands must be incorrect.

1.23 50 per cent of the imports of barley into a country come from the
Dominions ; 80 per cent of the total imports go to brewing ; 75 per cent
of the imports are grown in the Noryjje^^HSeimsphere ; 80 per cent of
Northern-grown barley goes to brewing ; 100 per cent of foreign Southern-
grown barley goes to stock-feeding. Show that the foreign Northern-

THEORY OF STATISTICS

grown barley which goes to brewing cannot be less than 30 per cent nor
more than 50 per cent of the total imports.

(It is assumed that brewing and stock-feeding are the only two uses to
w^hich imported barley is put.)

1 .24 A penny is tossed three times and the results, heads and tails, noted.
The process is continued until there are 100 sets of threes. In 69 cases
heads fell hrst, in 49 cases heads fell second, and m 53 cases heads fell
third. In 33 cases heads fell both first and second, and in 21 cases heads
fell both second and third. Show that there must have been at least 5
occasions on which heads fell three times, and that there could not have
been more than 15 occasions on which tails fell three times, though there
need not have been any.

CHAPTER TWO

ASSOCIATION OF ATTRIBUTES

Independence

2.1 If there is no sort of relationship of any kind between two attributes
A and B, we expect to find the same proportion of amongst the B's
as amongst the not-i5’s. We may anticipate, for instance, the same
proportion of abnormally wet seasons in leap years as in ordinary years,
the same proportion of male to total births when the moon is waxing as
when it is waning, the same proportion of heads whether a coin be tossed
with the right hand or the left.

Two such unrelated attributes may be termed independent, and we
have accordingly as the criterion of independence for A and B —

(^5),

. . .

{B)

If this relation holds good, the corresponding relations

(aB)_

Jap)

(B)'

(AB)_

(A)

(a)

{A)

(a)

must also hold. For it follows at once from (2.1) that

_{P)-(AP)

(B)

that is,

{aB)_

Jap)

(B) ■

and the other two identities may be similarly deduced.

The student may find it easier to grasp the nature of the relations stated
if the frequencies are supposed grouped into a table with two rows and two
columns, thus —

THEORY OF STATISTICS

Attribute

Total

(AB)

(«B)

(a)

Total

(B)

Equation (2.1) states a certain equality for the columns ; if this holds
good, the corresponding equation

(A) - (a)

must hold for the rows, and so on.

Forms of the criterion of independence
2.2 The criterion may, however, be put into a somewhat different
and theoretically more convenient form. The equation (2.1) expresses
(AB) in terms of (B), (fi) and a second-order frequency (Afi) ; eliminating
this second-order frequency we have —

{AB)_ {AB)+{A^) _{A)
(B) (B)+m N

i.e. ?n words, the proportion of A's amongst the B's is the same as in the
population at large.” The student should learn to recognise this equation
at sight in any of the forms —

{AB)_{A)

{B) N
(AB)_{B)

(A) N

N N' N

(<z)>

(*)

(c)

(d) J

( 2 . 2 )

The equation (d) gives the important fundamental rule : If the attributes
A and B are independent, the proportion of AB*s in the population is equal
to the proportion of .4's multiplied by the proportion of
The advantage of the forms (2.2) over the form (2.1) is that they give
expressions for the second-order frequency in terms of the frequencies of
the first order and the whole number of observations alone ; the form
(2.1) does not.

ASSOCIATION OF ATTRIBUTES

Example 2.1. — If there are 144 A's and 384 B's in 1024 observations,
how many AB’s will there be, A and B being independent ?

144x384

1024

=54

There will therefore be 54 AB's.

Example 2.2.— If the ^4's are 60 per cent, the B's 35 per cent, of the
whole number of observations, what must be the percentage of AB*s in
order that we may conclude that A and B are independent ^

60x35

100

=21

and therefore there must be 21 per cent (more or less closely, cf. 2.8 and
2.9 below) of in the population to justify the conclusion that A and
B are independent.

2.3 It follows from 2.1 that if the relation (2.2) holds for any one of the
four second-order frequencies, e.g. [AB), similar relations must hold for
the remaining three. Thus we have directly from (2.1) —

giving

{Ap)_ [AB)A^{Ap) __{A)

iP) (B)+{^) N

.{Am

and so on. This is seen at once to be true on consideration of the fourfold
table on page 20. For if (AS) takes the value {A){B) jN, (A.p) must take
the value {A)(P) jN to keep the total of the row equal to (A), and so
on for the other rows and columns. The fourfold table in the case of
independence must in fact have the form —

Attribute '

Total

{Am IN

(Am IN

(A)

{=){B)IN

(am IN

(a)

Total

(■B)

(A)

Example 2.3. — In Example 2.1 above, what v^ould be the number of
afi's, A and B being independent ?

(a) =1024-144 =880
(/?) =1024 -384 =640
, , 880 x 640

1024

THEORY OF STATISTICS

2A Finally, the criterion of independence may be expressed in yet a
third form viz. in terms of the second-order frequencies alone. If A and
B are independent, it follows at once from the preceding section that

And evidently {uB){Afi) is equal to the same fraction
Therefore

{AB){oc^)=^{ocB){A/3}

(AB) ^

(aB) {ap)

{AB) _ {a.B)

[AP) ~ [ap)

The equation (6) may be read : The ratio of ri's to a’s amongst the
B’s is equal to the ratio of A's to a’s amongst the and (c) similarly.
This form of criterion is a convenient one if all the four second-order
frequencies are given, enabling one to recognise almost at a glance whether
or not the two attributes are independent.

Example 2,4. — If the second-order frequencies have the following values,
are A and B independent or not ^

{AB)=^A10 (aB)=90 (ri^)=290

Clearly

{AB){ap) > (uB){AP)
so A and B arfe not independent.

(a)

(b)

(c)

(2.3)

Association

2S Suppose now that A and B are not independent, but related in some
way or other, however complicated.

Then if

{AB)>

mB)

A and B are said to be positively associated, or sometimes simply associated.
If, on the other hand,

{AB)<

{ A){B)

A and B are said to be negatively associated or, more briefly, disassociated.
The student should carefuUy note that in statistics the word
“ association ” has a technical meaning different from the one current in
ordinary speech. In common language one speaks of A and B as being
“ associated ” if they appear together in a number of cases. But in

ASSOCIATION OF ATTRIBUTES

statistics A and B are associated only if they appear together m a greater
number of cases than is to be expected if they are independent. Thus,
if we consider means of land transport as dichotomised into road and rail
travel, we may say, in the customary use of the term, that road transport
is associated with speed. But it does not follow that the two aie statisti-
cally associated, because rail transport may equally be associated with
speed and, in fact the attribute speed may be independent of the means
of travel in these two manners.

Association, therefore, cannot be inferred from the mere fact that some
are B*s, however great the proportion ; this principle is fundamental
and should always be borne in mind.

Complete association and disassociation

2.6 We have now to consider in what circumstances we may regard
the association of two attributes as complete. Tw'o courses are open to
us. Either we may say that for complete association all A's must be
B's and all B's must be ^'s, m which case it must follow that the
and the B's occur in the population m equal numbers ; or we may adopt
a rather wider meaning and say that all A's are B's or all S's are A's,
according to whether the ^’s or the B*s are in the minority. Similarly,
complete disassociation may be taken either as the case when no A*s are
B*s and no a's are /?*s, or more widely as the case when either of these
statements is true.

We shall adopt the wider definition In the sequel. Thus two attributes
are completely associated if one oi4hem cannot occur without the other,
though the other may occur without the one.

Measurement of intensity of association

2.7 It follows from the foregoing that if two attributes are completely
associated, [AB) must be equal to (.4) or (B), whichever is the smaller.
If they are completely disassociated, (AB) must be equal to zero
or to (A) + (B)--N whichever is the greater. (AB) must in general lie
between these two limits. We may thus regard the divergence of (AB)
from the “independence*' value {A)(B) jN towards the limiting value
in either direction as indicating the intensity of association or disassociation,
so that we may speak of attributes as being more or less, highly or slightly,
associated. This conception of degrees of association quantitatively
expressible is important, and we return in a later section to consider^the
formulae which may be used to measure such degrees.

Sampling fluctuations

2.8 When the association is very slight, i.e. where (^J3) differs from
{i4)(B) /JV* by only a few units or by a small proportion, it may be that
such association is not really significant of any definite relationship. To
give an illustration, suppose that a coin is tossed a number of times, and.

THEORY OF STATISTICS

the tosses noted in pairs ; then 100 pairs may give such results as the
following (taken from an actual record) —

First toss heads and second heads
„ „ „ tails

First toss tails and second heads
.. .. tails

If we use A to denote heads in the first toss, B “ heads in
the second, we have from the above (^)=44, (B)=53. Hence

(A){B) /IV’==:^^~^=23-32, while actuaUy {AB) is 26. Hence there is a

positive association, in the given record, between the result of the first
throw and the result of the second. But it is fairly certain, from the
nature of the case, that such association cannot indicate any real con-
nection between the results of the two throws ; it must therefore be due
merely to such a complex system of causes, impossible to analyse, as leads,
for example, to differences between small samples drawn from the same
matenal. The conclusion is confirmed by the fact that, of a number of
such records, some give a positive association (like the above), but others
a negative association.

2.9 An event due, like the above occurrence of positive association, to
an extremely complex system of causes of the general nature of which
we are aware, but of the detailed operation of which we are ignorant, is
sometimes said to be due to chance, or better to the chances or fluctuations
of sampling.

A little consideration will suggest that such associations due to the
fluctuations of sampling must be met with in all classes of statistics. To
quote, for instance, from 2.1, two illustrations there given of independent
attributes, we know that in any actual record we should not be hkely to
find exactly the same proportion of abnormally wet seasons in leap years
as in ordinary years, or exactly the same proportion of male births when
the moon is waxing as when it is waning. But so long as the divergence
from independence is not well marked we must regard such attributes
as practically independent, or dependence as at least unproved.

The discussion of the question, how great the divergence must be
before we can consider it as ** well marked,'" must be postponed to the
chapters dealing with the theory of sampling. At present the attention
of the student can only be directed to the existence of the difficulty, and
to the serious risk of interpreting a " chance association " as physically
significant.

The choice of a suitable form for testing association

2.10 The definition of 2.5 suggests that we are to test the existence

or the intensity of association between two attributes by a comparison

ASSOCIATION OF ATTRIBUTES

of the actual value of (AB) with its independence value (as it may be
termed) {A){B)IN. The procedure is from the theoretical standpoint
perhaps the most natural, but it is more usual, and is simplest and best
iri practice, to compare proportions, e.g. the proportion of A *s amongst the
with the proportion amongst the /?’s. Such proportions are usually
expressed in the form of percentages or proportions per thousand.

It will be evident from 2.1 and 2.2 that a large number of such com-
parisons are available for the purpose, and the question arises, therefore,
which is the best comparison to adopt ?

2.11 Two principles should decide this point : (1) of any two comparisons,
that is the better which* brings out the more clearly the degree of associa-
tion ; (2) of any two comparisons, that is the better which illustrates the
more important aspect of the problem under discussion.

The first condition at once suggests that comparisons of the form

{AB) {Afi)

(B) (A)

(2.4)

are better than comparisons of the form

{AB) (Jl
(B) N

(2.5)

For it is evident that if most of the objects or individuals in the population
are B*s, i.e. if {B) /N approaches unity, (AB) j{B) will necessarily approach
(A) jN even though the difference between (AB)l(B) and is

considerable. The second form of comparison may therefore be mis-
leading.

Setting aside, then, comparisons of the general form (2.5), the question
remains whether to apply the comparison of the form (2.4) to the rows or
the columns of the table, if the data are tabulated as on page 21. This
question must be decided with reference to the second principle, i.e. with
regard to the more important aspect of the problem under discussion,
the exact question to be answered, or the hypothesis to be tested, as
illustrated by the examples below. Where no definite question has to be
answered or hypothesis tested both pairs of proportions may be tabulated.

Example 2.5, — Association between inoculation against cholera and
exemption from attack. (Data from Greenwood and Yule, Proc. Roy,
Soc, Med., 1915, 8, 221, T?ible III).

Not attacked

Attacked

Total

Inoculated

276

279

Kot inoculated .

473

539

Total

749

818

THEORY OF STATISTICS

Here the important question is, How far does inoculation protect from
attack ? The most natural comparison is therefore —

Percentage of inoculated who were not attacked • • 98*9

„ not inoculated „ „ • * 87*8

Or we might tabulate the complementary proportions —

Percentage of inoculated who were attacked * • 1*1

,, not inoculated „ ,, * • 12*2

Either comparison brings out simply and clearly the fact that inocula-
tion and exemption from attack are positively associated {inoculation and
attack negatively associated).

We are making above a comparison by rows in the notation of the table
on page 21, comparing [AB] /{A) with (aB) /(a), or (A^) j{A) with (a/?) /(a).
A comparison by columns, e.g. (AB)I{B) with (Afi) /{jS), would serve
equally to indicate whether there was any appreciable association, but
would not answer directly the particular question we have in mind —

Percentage of not- attacked who were inoculated * • 36*8

„ attacked „ „ * * 4*3

Example 2.6. — Eye-colour of father and son (material due to Gallon,
as given by Pearson, PML Trans., A, 1900, 195, 138 ; the classes 1, 2 and
3 of the memoir treated as “ light ”).

Fathers with light eyes and sons with light eyes {AB) ^ • 471

>, „ , „ „ not light „ {Ap) • • 151

,, not hght „ „ light „ {ccB) * • 148

„ „ „ not light (a/?) • • 230

Required to find whether the colour of the son's eyes is associated with
that of the father's. In cases of this kind the father is reckoned once for
each son ; e.g. a family in which the father was light-eyed, two sons light-
eyed and one not,* would be reckoned as giving two to the class AB and
one to the class Ap.

The best comparison here is —

Percentage of light-eyed amongst the
of light-eyed fathers

Percentage of light-eyed amongst the
of not-light-eyed fathers *

But the following is equally valid —

Percentage of light-eyed amongst
fathers of light-eyed sons

Percentage of light-eyed amongst
fathers of not-light-eyed sons

sons ^

sons

76 per cent
39

the 1 .

Y 76 per cent

}

ASSOCIATION OF ATTRIBUTES 'Z^

The reason why the former comparison is preferable is that we usually
wish to estimate the character of offspring from that of the parents, and
not vice versa. Both modes of statement, however, indicate equally
clearly that there is considerable resemblance between father and son.

Example 2 7 — Association between inoculation against cholera and
exemption from attack, five separate epidemics (cf. Example 2.5, data
from Tables IX, X, XXVIII, XXIX, XXXI of the paper there cited.)

Not attacked

Attacked

Total

Inoculated

192

196

Not inoculated *

113

147

Total •

305

343

Not attacked

Attacked

Total

Inoculated

5.751

5,778

Not inoculated •

6,351

198

6,549

Total •

12,102

225

12,327

Not attacked

Attacked

Total

Inoculated

4,087

4,092

Not inoculated *

113,856

1,144

115,000

Total •

117,943

1,149

119,092

Not attacked

Attacked

Total

Inoculated

8,332

8,340

Not inoculated •

84,444

556

85,000

Total *

' • 92,776

564

93,340

Not attacked

Attacked

Total

Inoculated

4,870

4,875

Not inoculated •

153,096

904

154,000

Total •

157,966

909

158,875

With the table of Example 2.5 the above give data for six separate
epidemics, in all of which the same method of inoculation appears to have
been used : the data refer to natives only, and the numbers of observations
are sufficiently large to reduce fluctuations of sampling " within reason-
ably narrow limits. The proportions not attacked are as follows —

THEORY OF STATISTICS

Proportion not attacV'd

Not inoculated

InocuUted

Diiference

1 •

• 0-8776

0-9892

0-1116

2 •

• 0-7687

0-9796

0-2109

3 •

- 0-9698

0-9953

0-0255

4 •

- 0-9901

0-9988

0-0087

5 •

• 0-9935

0-9990

0-0055

6 •

•

• 0-9941

0-9990

0-0049

In each case inoculation and exemption from attack are positively
associated, but it will be seen that the several proportions, and the differ*
ences between them, vary considerably. Evidently m a very mild
epidemic this difference can only be small, and the question arises how
far the data for the separate epidemics can be said to be consistent in
their indication of the efficiency of the inoculation. This is not a
simple question to answer : the more advanced student is referred to the
discussion in the original.

The symbols {AB)q and d

2.12 The values that the four second-order frequencies take in the
case of independence, viz.

{Am wi

N ' N ' N ' N

are of such great theoretical importance, and of so much use as reference-
values for comparing with the actual values of the frequencies (AB), (ocB),
[AJ]) and (a/?), that it is often desirable to employ single symbols to denote
them. We shall use the symbols

mB)

{oc^)q —

(«)(/g)

{o^B),

MB)

mfi)

If 6 denote the excess of {AB) over (^B)o, then, in order to keep the totals
of rows and columns constant, the general table (cf. the table for the case
of independence on page 21) must be of the form —

Attribute

Total

{AB)^-\rd

(A)

(a)

Total

{^)

Therefore, quite generally we have —

(^B) -Ca/)o= W)o- W ) =8

ASSOCIATION OF ATTRIBUTES 29

2.13 The value of this common difference S may be expressed in a form
that is useful to note. We have by definition —

Bring the terms on the right to a common denominator, and express all
the frequencies of the numerator in terms of those of the second order ;
then we have —

1 r {AB)l(AB)+{aB) + iA^) + (oc/S)] *1

- [{A B) +{AM(AB) + (ocB)] J

=l-{{AB)i^^)-{aB){A^)}

That is to say, the common difference is equal to 1 /Nth of the difference
of the ** cross-products {AB)(afi) and

It is evident that the difference of the cross-products may be very
large if N be large, although S is really very small In using the difference
of the cross-products to test mentally the sign of the association in a case
where all the four second-order frequencies are given, this should be
remembered ; the difference should be compared with N, or it will be
liable to suggest a higher degree of association than actually exists.

Example 2.8 — The following data were observed for hybrids of Datura
(Bateson and Saunders, Report to the Evolution Committee of the Royal
Society, 1902) —

Flowers violet, fruits prickly (AB)

„ „ smooth (A/3)

Flowers white, „ prickly (ocB)

„ „ smooth (ccj3)

Investigate the association between colour of flower and character of
fruit.

Since 3x47=141, 12x21=252 i.e. (AB)(a^) < {ccB){Ap), there is
clearly a negative association; 252—141=111, and at first sight this
considerable difference is apt to suggest a considerable disassociation. But
^=111/83=1*3 only, and forms a small proportion of the frequency, so
that in point of fact the disassociation is small, so small that no stress can
be laid on it as indicating anything but a fluctuation of sampling. Work-
ing out the percentages we have —

Percentage of violet-flowered plants with \
prickly fruits • • • *

Percentage of white-flowered plants with
prickly fruits ^ •

80 per cent

THEORY OF STATISTICS

Coefficient of association

2.14 In the previous examples we have judged the association by
comparing the class-frequencies with those which would exist if the data
were given by independent attributes, and we can form a rough idea of
the strength of the association by examining the extent of the difference.
This is sufficient for almost all practical purposes, although, if the data
are likely to be affected seriously by fluctuations of random sampling,
some test of the significance of the difference is also necessary. Apart
from this question, however, it is sometimes convenient to measure the
intensities of the associations by means of a coefficient.

It is dearly convenient if such a coefficient can be devised as to be
zero if the attributes are independent, -f 1 if they are completely associated
and— 1 if they are completely disassociated.

2.15 Many such coefficients may be devised, but perhaps the simplest
possible (though not necessarily the most advantageous) is the expression —

{AB){oc^)+{A^){aB)

wliere S is the symbol used in 2,12 and 2.13 for the difference (AB) —
(AB)q. It IS evident that Q is zero when the attnbutes are independent,
for then S is zero : it takes the value ^-l when there is complete association,
for then the second term in both numerator and denominator of the
first form of the expression is zero : similarly it is —1 where there is
complete disassociation, for then the first term in both numerator and
denominator is zero. Q may accordingly be termed a coefficient of
association. As illustrations of the values it will take in certain cases,
the association between light eye-colour in father and in son (Example 2.6)
is -t-0-66; between coldur of flower and prickliness of fruit in Datura
(Example 2.8), — O' 28: a disassociation which, however, as ailready
stated, is probably of no practical significance and due to mere fluctuations
of sampling.

The student should note that if all the terms containing A are multiplied
by a constant, the value of Q is unaltered. Similarly for a, B and fi.
Hence Q is independent of the relative proportions of A’s and a’s in the
data. This property is important, and renders such a measure of associa-
tion specially adapted to cases in which the proportions are arbitrary
(e.g. experiments).

ASSOCIATION OF ATTRIBUTES

3 ^

2.16 Another coefficient which has the same property is the coefficimt
of colligation.

It is easy to show that

^ (AB){af)
{AB)(ocfi)

( 2 . 6 )

1+Y^

(2.7)

Association in sub-populations

2.17 Up to this point we have considered association between two
attributes in a population without regard to whether any information
existed about other attributes in the population. If, however, such
information does exist and, say, we can find the frequency-classes of
attributes C, D, etc., the question anses, What are the associations of
A and B in the sub-populations C, y, CD, etc.?

Thus, if A=standard of health and D=consumption of food, the fore-
going discussion would enable us to examine whether health and food-
consumption were associated in any particular population, say the popula-
tion of Great Bntain. But we might want to go further than this and
examine the association between A and B among males, or among the
poorer classes, and compare it with the association among females or among
the well-to-do classes, respectively. Defining C=males and D=poor, this
amounts to examining the associations of A and B in the populations C, y,
D and d.

2.18 Associations of this kind are of the utmost importance in statistical
practice. As instances of the ways in which they arise let us consider the
following two illustrations —

(1) Suppose that we have established, in the manner of foregoing
sections, a positive association between inoculation and exemption from
smallpox in a population of persons. It is natural to infer that this associa-
tion is due to some causal relation between the two attributes and may be
expected to recur in the future ; in short, that smallpox is prevented by
vaccination.

This rather hasty conclusion might, however, meet an opponent who
argues in this way : vaccination is accepted among the well-to-do classes,
but is looked on with suspicion by the lower classes. For this and other
reasons most of the unvaccinated persons arc drawn from the lower classes.
But these are precisely the people whom, from the unhygienic conditions
under which they live, one would expect to be exposed to infection and
who, moreover, being malnourished, would be more likely to contract
disease when they were infected. Hence the comparative exemption of

THEORY OF STATISTICS

the vaccinated persons is not due to the fact that they have been vaccinated,
but to the fact that they belong to the well-to-do classes. It is, as it were,
an accident that these people also happen to be from a class which favours
vaccination.

Denoting vaccination by A, exemption from attack by B and hygienic
conditions by C, this argument amounts to sa 5 dng that the observed
association between A and B is not of itself causally direct, but is due to
the associations of both A and B with C.

Now it is clear that this objection could not be lodged if the hygienic
conditions among all the members of the population were the same. If,
therefore, we examine the association of A and B in the sub-population C
and still find an association, the supposed argument will be refuted. We
are thus led to a consideration of the association in that sub-population.

(2) As a second example, suppose that an association is noted between
the presence of an attribute in the father and the presence in the son, and
also between the presence in the grandfather and the presence in the grand-
son. The question which arises here is : Does the resemblance between
grandfather and grandson arise from a kind of hereditary transmission
which may, in the common phrase, skip a generation,” or is it merely
due to the fact that the grandfather is like the father and the father is like
the son ?

Denoting the presence of the attribute in the son, father and grand-
father by A, B and C, the question is : Is the association between A and C
due to associations between A and JS, and B and C ?

If the association between A and C is observed among all the cases in
which the father possesses the attribute or all those in which he does not,
and is still sensible, clearly the association between A and C cannot be due
to associations between A and B, B and C ; hence, as before, to resolve
the question we are led to consider the association between A and C in the
sub-populations B and

2.19 Generally, ambiguity of the type to which we have just referred
arises from the fact that the population under discussion contains not
merely objects possessing the third attribute alone, but a mixture of
objects with and without it. To meet the requirements of the discussion
we have to consider the associations in sub-populations wherein this attri-
bute is entirely absent or entirely present. By this means we can go
deeper into the nature of the underlying causes and eliminate certain
possible explanations of the type : an association between A and B does
not mean that the two are directly related, but only that each is associated
with a third attribute C.

Partial associations

2.20 The associations between A and B in sub-populations are called
partial associations, to distinguish them from the total associations between
A and B in the population at large:

ASSOCIATION OF ATTRIBUTES

As for total association, A and B are said to be positively associated
in the population of C’s if

(ABC) >

{AC)(BC)

and negatively associated in the converse case.

Similarly they are positively associated in the population of CD’s if

. . . (. 9 )

and so on. These formulae are derived from the formula for total associa-
tion by specif 5 ang the population in which the partial association exists.

Alternative forms of the conditions for partial association
2,21 As in the case of total association, the above forms can be written
in many ways, adapted to the nature of the data and of the question
which is to be answered. The partial association is most conveniently
tested by comparisons of percentages or proportions in the manner of 2,2,
and we may quote the four most convenient comparisons in the case
of three attributes —

(ABC) ^ {AC)
(BC) ^ (C)
{ABC) (AfiC)
(BC) ^ (/?C)

(a)

(ABC) (BC)
(AC) (C)
(ABC) ^ (ccBC)
(AC) ^ (aC)

• (&)'

- ( 2 . 10 )

• w

Similar formulae may be written down for the cases of four or more
attributes, and the methods of this chapter are applicable to such cases.
For the sake of simplicity we shall, however, confine ourselves to three
attributes hereafter.

Example 2.9. — The following are the proportions per 10,000 of boys
observed with certain classes of defects amongst a number of school*
children. (A) denotes the number with development defects, [B] the
number with nerve signs (D) the number of the “ dull."

N 10,000 {AB) 338

{A) 877 (AD) 338

{B) 1,086 (BD) 455

(D) 789 (ABD) 153

The Report (referred to in Example 1.1) from which the figures are drawn
concludes that " the connecting link between defects of body and mental
dullness is the coincident defect of brain which may be known by observa-
tion of abnormal nerve signs." Discuss this conclusion.

The phrase “ connecting link " is a little vague, but it may mean that
the mental defects indicated by nerve signs B may give rise to develop-

THEORY OF STATISTICS

ment defects A, and also to mental dullness D ; A and D being thus
common effects of the same cause B (or another attribute necessarily
indicated by B) and not directly influencing each other. The case is
thus similar to that of the first illustration of 2.18 (liability to smallpox
and to non- vaccination being held to be common effects of the same
circumstances), and may be similarly treated by investigation of the
partial associations between A and D for the populations B and As the
ratios {A)IN, {B)/N, {D) jN are small, comparisons of the form (2.10),
{a) and (6) above, may be used.

The following ffgures illustrate, then, the association between A and D
for the whole population, the S-population and the y6^-population —

For the entire material —

789

Proportion of the dull==(Z)) (N • • = — — 7*9 per cent

„ ,, defectively developed who'll 338 ^

weredull=(4D)/(^) • • -J” 8^ ”

For those exhibiting nerve signs —

4SS

Proportion of the dull~(-BP) l{B) • • = — ^ =41 *9 per cent

1,086

„ ,, defectively developed who 1 __ 153 -

werediill=(45D)/(^i5) • • • 1 ^

For those not exhibiting nerve signs —

Proportion of the dull=(ySZ?) /(/ff) •

,, „ defectively developed who

were duU=(J^y^D) /{Afi)

= 3*7

The results are extremely striking ; the association between A and D
is high both for the material as a whole (the population at large) and for
those not exhibiting nerve signs (the ^-population), but it is small for those
who do exhibit nerve signs (the jB-population).

This result does not appear to be in accord with the conclusion of the
Report, as we have interpreted it, for the association between A and D
in the /^-population should in that case have been low instead of high.

Notation for partial associations

2.22 We now introduce a notation which is analogous to that used
for total associations. It will be remembered that in 2.13 we wrote —

{AB),^

{Am

d^{AB)--{AB)^

ASSOCIATION OF ATTRIBUTES

We now write—

(Q ’ ^ (CD) . (2.11)

SAB.c = iABC)-{AB.C)^. SAB.Ci>={ABCD)-{AB .CD)o, etc )

The <^-nnmbers measure the divergence of the actual frequencies from
those which would exist if the attributes were independent in the sub-
population under discussion

It IS also possible to generalise the coefficient of association Q by defining
partial coefficients of the type

{ABC)(a^C)^{ApC){aBC) ^

^ • (ABC)(afiC)+(A^C){aBC)\

{QSab.c I ’ * ‘

~(ABC){afiC)+{A^C){ocBC) ^

The student will notice that the formulae for the ^-nunibers and for
the Q numbers are obtained from the expressions for total association by
specifying the population in which the partial association is to be con-
sidered. They need not therefore be memorised.

Number of partial associations

2,23 For three attributes A, B, C there are three total associations,
namely, those of A with B, B with C and C with A , and six partial
associations, namely, those of A and B in C and y, B and C in ^ and a,
and C and ^ in ^ and

For four attributes there are fifty-four associations ; for we can choose
tw^o attributes from four in six ways, and there are nine associations for
each pair (one total, four partials in the sub-populations specified by one
attribute, and four partials in the sub-populations specified by two).

We state without proof that for
n{n—l)

n attributes there are

n{n—l)

3«-'

associations. Of these,

are total and the remainder partial. For

n > 4 this number is so large as to be almost unmanageable. For instance,
if ft =5 it is 270, and if ft =6 it is 1215.

The large number of partial associations which exists might be thought
to occasion some difficulty. We may, however, reassure ourselves by
two considerations.

In the first place, it is rarely necessary to investigate in any practical
instance all the partial associations which are theoretically possible. For
instance, in Example 2.9 the total and partial associations between A
and D were alone investigated ; those between A and B, B and D were
not essential for answering the question which was asked.

THEORY OF STATISTICS

Relations between partial associations

2.24 In the second place, a theoretical discussion of the partial associa-

fl(fl 1 \

tions IS assisted by the following result : The — associations

Jmt

are

all expressible in terms of 2”~(n + l) algebraically independent associa-
tions, together with the class-frequencies iV, {A), {B), (C), etc.

In fact, we saw in Chapter 1 that all the class-frequencies can be
expressed in terms of the positive class-frequencies, which are 2” in
number in the case of n attributes. Hence the frequencies N, (A), (B),
(C), etc., of which there are {n+\), together with the 2»— (w + l) other
positive frequencies, completely determine the data, and hence determine
the associations, which are expressed in terms of the data. Hence the
number of algebraically independent associations which can be derived
is only (w+1).

2.25 In practice the existence of these relations is of little or no value.
The formal relations between the ratios and the {^-numbers which express
the associations are, in fact, so complex that lengthy algebraic manipula-
tion is necessary to express those which are not known in terms of those
which are. It is usually better to evaluate the class-frequencies and
calculate the desired results directly from them.

2.26 There is, however, one result which has important theoretical
consequences.

We have, by definition,

[y)

Hence,

B^y-

<AB)

={AB)

I (AC)(BC)(y)+(Ay)(By)(C) [
r-J— m(AC)(BC)-(A)fCUBn-(R

-^|I\r(^C)(BC)-{4)(C)(BC)-(B)(C)(4C)

+(^){B)(C);

This gives us the sum of the ^-numbers for the partial associations of A
and B in C and y in terms of the total associations between A. B and C.

ASSOCIATION OF ATTRIBUTES

Now suppose that A and B are independent in C and y. Then we
have —

and

^ab=^~.-—SacSbc

(Q(y)

Sab is not zero unless one or both of Sac, Sbc are zero.

Hence, if A and B are independent within the populations of C’s and
not-C’s, they will nevertheless -be associated in the population at large
unless C is independent of ^ or or both. -

Illusory associations

2.27 This peculiar result indicates that, although a set of attributes
independent of A and B will not afiect the association between them, the
existence of an attribute C with which they are both associated may give
an association in the population at large which is illusory in the sense that
it does not correspond to any real relationship between them. If the
associations between A and C, B and C are of the same sign, the resulting
association between A and B will be positive; if of opposite signs,
negative.

The cases which we discussed at the beginning of this chapter are
instances in point. In the first illustration we saw that it was possible to
argue that the positive associations between vaccination and hygienic con^-
ditions, exemption from attack and hygienic conditions, led to an illusory
association between vaccination and exemption from attack. Similarly, the
question was raised whether the positive association between grandfather
and grandchild may not be due to the positive associations between grand-
father and father, and father and child.

2.28 Misleading associations may easily arise through the mingling
of records which a careful worker would keep distinct.

Take the following case,, for example. Suppose there have been 200
patients in a hospital, 100 males and 100 females, suffering from some
disease. * Suppose, further, that the death-rate for males (the case mor-
tality) has been 30 per cent, for females 60 per cent. A new treatment is
tried on 80 per cent of the males and 40 per cent of the females, and the
results published without distinction of sex. The three attributes, with
the relations of which we are here concerned, are death, treatment and male
sex. The data show that more males were treated than females, and more
females died than males ; therefore the first attribute is associated nega-
tively, the second positively, with the third. It follows that there will be
an illusory negative association between the first two — death and treatment.

THEORY OF STATISTICS

If the treatment were completely inefficient we should, m fact, have the

following results —

Treated and died *

„ and did not die .
!Not treated and died

,, and did not die

Males

Females

Total

i.e. of the treated, only 48/120=40 per cent died, while of those not
treated 42 /80=52-5 per cent died. If this result were stated without any
reference to the fact of the mixture of the sexes, to the different proportions
of the two that were treated and to the different death-rates under normal
treatment, then some value in the new treatment would appear to be
suggested To make a fair return, either the results for the two sexes
should be stated separately, or the same proportion of the two sexes must
receive the experimental treatment. Further, care would have to be taken
in such a case to see that there was no selection (perhaps unconscious) of
the less severe cases for treatment, thus introducing another source of
fallacy [death positively associated with severity, treatment negatively
associated with severity, giving rise to illusory negative association between
treatment and death).

2.29 Illusory associations may also arise in a different way through
the personality of the observer or observers. If the observer’s attention
fluctuates, he may be more likely to notice the presence of A when he
notices the presence of B, and vice versa ; in such a case A and B (so far as
the record goes) will both be associated with the observer’s attention C,
and consequently an illusory association will be created. Again, if the
attributes are not well defined, one observer may be more generous than
another in deciding when to record the presence of A and also the presence
of B, and even one observer may fluctuate in the generosity of his marking.
In this case the recording of A and the recording of B will both be associated
with the generosity of the observer in recording their presence, C, and an
illusory association between A and B will consequently arise, as before.

Determination of sign of association when the data are incomplete

2.30 It is important to notice that, though we cannot actually determine
the partial associations unless the third-order frequency (ABC) is given,
we can make some conjecture as to their signs from the values of the
second-order frequencies.

In 2.26 we have —

. . ( 2 ,„,

(^) (7)

Hence, if the expression on the right is positive, one at least of (Jar.c,
SdB.y, is positive, i.e. A and B are positively associated either in C or y
or both. Similarly, if the expression is negative, A and B are negatively

ASSOCIATION OF ATTRIBUTES

associated either in C or in y or in both. Finally, if the expression is
zero, A and B are either independent in both C and y, or positively
associated in one and negatively in the other.

The expression may be thrown into a form more convenient when
percentages are given. Dividing through by (B) we have —

SABC+SAB.y _ (AB} (AC) (BC) _ GItO (M f2 IS)

(B) (B) (C) (B) (y) (B) ’

The following example illustrates the method.

Example 2.10 (Figures compiled from the Registrar-GeneraV s Decennial
Supplement, 1931, Part II a — 1938). The following are the mean annual
death-rates for occupied (including retired) males of 16 years of age and
over for England and Wales during the three years 1930-1932.

Death rate per thousand

Occupied and retired males over 16 . . 14*63

Farmers over 16 19*68

Anglican clergy over 16 . . .27*81

Coal hewers and getters over 16 . . 14*69

At first sight it appears that coal hewing is about the average in healthiness
(as measured by death rate) and that farmers and clergy are decidedly
unhealthy. These conclusions are quite wrong.

The following are the proportions of the occupations 65 years old or
more at the census date 1931 —

Proportion per thousand
65 years of age or more

Occupied and retired males . . .86*8

Farmers 172*1

Anglican clergy ... . 279*4

Coal hewers and getters . . .68*6

For the whole class of occupied and retired males the death rates for the
groups 1^65 years and- 65 years and over were 7*93 per thousand and
85*10 per thousand.

If A denote death, B the given occupation, C old age, we have to apply
the principles of equation (2.15), calculate what would be the death-rate
for each occupation on the supposition that the rates for occupied and
retired males in general (7*93 and 85*10) apply to each of the separate
age-groups (16-65, 65 and over), and see whether the total death-rate
so calculated exceeds or falls short of the actual death-rate. If it exceeds
the actual rate the occupation must on the whole be healthy ; in the
contrary case, unhealthy. Thus we have the following calculated death
rates —

Farmers .... 7*93x ‘8279-h85* 10 X *1721=21*20

Anglican clergy . 7*93x •7206-f85*10x *2794 =29*48

Coal hewers and getters * 7*93x *9314-4-85* lOx *0686=13*21

THEORY OF STATISTICS

The calculated rate for farmers and clergy largely exceeds the actual
rate ; these occupations then must, on the whole, be healthy. On the
other hand the rate for coal hewers and getters falls short of the actual
rate and this occupation is relatively unhealthy. The true facts are
masked in the death-rates for the occupations taken irrespective of age by
the various proportions of young and old engaged in the occupations.

It is evident that age-distributions vary so largely from one occupation
to another that total death-rates are liable to be very misleading. Similar
f^acies are liable to occur in comparisons of local death-rates, owing
to variations not only in the relative proportions of the old, but also in
the relative proportions of the two sexes.

It is hardly necessary to observe that as age is a variable quantity, the
above procedure for calculating the comparative death-rates is extremely
rough. The death-rate of those engaged in any occupation depends not
only on the mere proportions over and under 65, but on the relative
numbers at every single year of age. The simpler procedure brings out,
however, better than a more complex one, the nature of the fallacy involved
in assuming that crude death-rates are measures of healthiness.

Complete independence

2,31 The particular case in which all the 2»-~(n+l) given associations
are zero is worth some special investigation.

It follows, in the first place, that all other possible associations must be
zero, i.e. that a state of complete independence, as we may term it, exists.
Suppose, for instance, that we are given —

{AB):

{Bcy.

. {Am

miQ

(AC):

(ABC)--

SAC){BC) _(Am{C)
(C) m

Then it follows at once that we have also-

r,r>r^..{AB){BC)_{AB){AC)

(ABC) ^

i.e. A and C are independent in the population of B% and B and C in the
population of A's. Again,

(ABy)==(AB)-(ABQJ^-^^Mi^

_ (A)(B)(r) _ (Ay)(By)
m (y)

Therefore A and B are independent in the population of y’s. Similarly, it
may be shown that A and C are independent in the population of /?'s, B and
C in the population of a^s.

In the next place it is evident from the above that relations of the
general form (to write the equation symmetrically)

ASSOCIATION OF ATTRIBUTES

(ABC)_(A) (5) ^
N N ^ N ' N

, (2.16)

must hold for every class-frequency. This relation is the general form of
the equation of independence (2.2) (d).

2.32 It must be noted, however, that (2.16) is not a criterion for the
complete independence oi A, B and C in the sense that the equation

{AB)_[A) [B)

N N ' N

is a criterion for the complete independence of A and B. If we are given
N, [A) and (B), and the last relation quoted holds good, we know that
similar relations must hold for {Afi), {aB) and (a^ff). If iV, (A), (B) and
(C) be given, however, and the equation (2.16) holds good, we can draw no
conclusion without further information ; the data are insufficient. There
are eight algebraically independent class-frequencies in the case of three
attributes, while N, {A), (B), (C) are only four : the equation (2.16) must
therefore be shown to hold good for four frequencies of the third order
before the conclusion can be drawn that it holds good for the remainder, i.e.
that a state of complete independence subsists. The direct verification of
this result is left for the student.

Quite generally, if iV, (A), (B), (C), ... be given, the relation

{ABC . ■ ■ ) _{A) (B) (Q

N N ' N' N ' ■ ■ ‘ f

must be shown to hold good for 2*'— («+l) of the nth order classes before it
may be assumed to hold good for the remainder. It is only because

2»-(w+l)=l

when «=2 that the relation

(AB)_^(A) (B)

N N ' N

may be treated as a criterion for the independence ut A and B. //all the
n {n > 2) attributes are completely independent, the relation (2.17) holds
good ; but it does not foUow that if the relation (2.17) holds good they are
all independent.

SUMMARY

1. Two attributes are independent if the proportion of A's among the
B*s is the same as the proportion among the not-B's.

2. This definition can be expressed symbolically in numerous forms, in

THEORY OF STATISTICS

terms of either first-order or second-order frequencies. The form in which
the data are given, and the question which is to be answered, determine
which form is to be employed in any particular case.

3. Attributes which are not independent are said to be positively
associated if

{AB)>

ijm

and negatively associated if

(ABX

mB)

4. The statistical meaning of the word association '' is different from
the meaning ascribed to it in ordinary language.

5. Before association may be said to indicate a definite relation between
the attributes, it is necessary to be satisfied that the divergence from
independence is not due to fluctuations of sampling.

6. The divergence of the actual frequency from the '' independence "
frequency is denoted by the symbol S, and hence

d^{AB)

mB)

The coefficient of association is defined by

It is zero if the attributes are independent, +1 if they are completely
associated and —1 if they are completely disassociated. There are,
however, other forms of coefficient more advantageous in certain cases.

8. The association of A and B in sub-populations of the type C, y, CD,
CDE, etc. is called a partial association.

9. If

{ABC) >

{^C){BC)

(C)

A and B are positively associated in C ; and if

(ABCK(.d^

(0)

A and B are negatively associated in C.

10. There are ^ ^^3 ”"^ associations in a population characterised by

n attributes, - of which are total and the remainder partial.

11. All the associations are expressible in terms of iV, (4), (B), (C),
etc., and 2'»~{ft+l) algebraically independent associations. These relations
have, however, only a theoretical value.

ASSOCIATION OF ATTRIBUTES

12. If ^ and B are independent within the population of C's they will
nevertheless be associated within the population at large, unless C is inde-
pendent of either A ov B or both.

13. In interpreting an association between A and B it must be remem-
bered that this may arise owing to associations of A with C and B with
C. To resolve this point it is necessary to consider the partial associations
of A and in C and y.

14. Complete independence of n attributes occurs if 2« — (n-f 1) algebraic-
ally independent associations and hence all associations are zero. In this
case

- {ABC . . . ) _{A) (B) (C)

N N N N ' ‘ ^

but this last condition is not sufficient for complete independence.

EXERCISES

2.1 At the census of England and Wales in 1901 there were (to the nearest
1,000) 15,729,000 males and 16,799,000 females ; 3,497 males were returned
as deaf-mutes from childhood, and 3,072 females.

State proportions exhibiting the association between deaf-mutism from
childhood and sex. How many of each sex for the same total number
would have been deaf-mutes if there had been no association ?

2.2 Show, as briefly as possible, whether A and B are independent,
positively associated or negatively associated in each of the following
cases —

{a) 5,000 (A) = 2,350 {B) =3,100 (AB) = 1,600

(b) (.4)= 490 {AB)=^ 294 ((x)= 570 (a5)= 380

2.3 (Figures derived from Darwin's Cross- and Self-Jertilisation of
Plants) The table below gives the numbers of plants of certain species
that were above or below the average height, stating separately those
that were derived from cross-fertilised and from self-fertilised parentage.
Investigate the association between height and cross-fertilisation of
parentage, and draw attention to any special points you notice.

Species

Parentage cross-fer-
tilised Height —

Parentage self-fer-
tilised. Height —

Above

average

Below

average

Above

average

Below

average

Ipomaea purpurea.

Petunia wolacea ....

Reseda lutea

Reseda odorata ....

Lobelia fulgens ....

THEORY OF STATISTICS

2.4 (Figures from same source as Example 2.6 ; classes 7 and 8 of the
memoir treated as dark.'') Investigate the association between darkness
of eye-colour in father and son from the following data —

Fathers with dark eyes and sons with dark eyes {AB) . 50

„ „ „ not-dark eyes {A^) . 79

Fathers with not-dark eyes and sons with dark eyes {aB) . 89

„ „ „ not-dark eyes (a^) , 782

Also tabulate for comparison the frequencies that would have been
observed had there been no heredity, i.e. the values of {AB)q, etc.

2.5 (Figures from same source as above.) Investigate the association
between eye-colour of husband and eye-colour of wife ('‘ assortative
mating ") from the data given below.

Husbands with light eyes and wives with light eyes [AB) . 309
„ „ „ not-hght eyes [A^] . 214

Husbands with not-light eyes and wives with light eyes {txB) . 132
„ „ „ not-light eyes (ay?) . 119

Also tabulate for comparison the frequencies that would have been
observed had there been strict independence between eye-colour of husband
and eye-colour of wife, i.e., the values of [AB)q, etc., as in Exercise 2.4.

2.6 (Figures from the Census of England and Wales, 1891, vol. 3: the
data cannot be regarded as trustworthy.) The figures given below show
the number of males in successive age-groups, together with the number
of the blind (.4), of the mentally deranged [B) and the blind mentally
deranged [AB), Trace the association between blindness and mental
derangement from childhood to old age, tabulating the proportions of
insane amongst the whole population and amongst the blind, and also
the association coefficient Q of 2.15. Give a short verbal statement of
your results.

15-

25-

35-

45-

65-

75 and
upwards

(^) i
\b)

i^B)

3,304,230

844

2,820

2,712,521

1,184

6,225

2,089,010

1,165

8,482

1,611,077

1,501

9,214

1,191,789

1,752

8,187

770,124

1,905

5,799

444,896

1,932

3,412

161,692
1,701
1,098
i 9

2.7 Show that if

{AB)^ (aB), (^/?), (a/?),

{AB)^ (ocB\

be two aggregates corresponding to the same values of (^), [B), (a) and (/?),

(iB), -(.4B),=(aB),-(aB),=(JA)2-(^>S)x=(«/^). -(«/?).

ASSOCIATION OF ATTRIBUTES

2.8 Show that if

S={AB)-{AB)^

{ABY+{afi)^-{aBY-{APY==[{A)-{a)]m*m']-^Wd

2.9 The existence of association may be tested either by comparison of
proportions (e.g. {AB)jiB) with [A8)I{B)), as in 2.10 and 2.11, or by the
value of d as in 2.12 and 2.13. Show that

{BmUAB) {Afi)\

N 1 {B) (A) J

_(AYa)nAB) (gg)]

N (a)J

2.10 Spence and Charles, in An Investigation into the Health and Nutrition
of Certain of the Children of Newcastle-on^Tyne between the Ages of One
and Five Years (City and Council of Newcastle-on-Tyne, February 1934),
compared two groups of children, one belonging to the professional classes,
125 in number, and the other belonging to the labouring classes, 124 in

number. They found the following results —

Poor Well-to-do

Children Children

Percent Per cent

Below normal weight ... 55 13

Above normal weight ... 11 48

Find the coefficient of association between the weight of the children and
their social status.

2.11 (Data from the Report on the Spahlinger Experiments in Northern
Ireland, 1931-1934, H.M. Stationery Office, 1935.) In experiments on
the immunisation of cattle from tuberculosis the following results were
secured —

Cattle

Treatment

Died of
tuberculosis or
very seriously
affected

UnaJffected or
only slightly
affected

Total

Inoculated with vaccine

Not inoculated or inoculated with
control media

Total .

(The cattle were first inoculated with protective vaccine and then
deliberately infected with serious quantities of tubercle germs.)

Find the coefficient of association between inoculation and exemption
from serious tuberculosis.

THEORY OF STATISTICS

2.12 Criticise the following argument : Nearly all the are J5's, and
therefore A and B must be associated,” and state what suppressed premises
would justify it in the following cases —

“ 99 per cent of the people who drink beer die before reaching 100 years
of age. Therefore drinking beer is bad for longevity.”

” 99 per cent of the members who voted for the Army Estimates were
military officers. Therefore it was unfair to suppose that the voting was
unbiased.”

'' In every country where the sale of contraceptives is tolerated by the
Government the birth-rate is declining. Therefore contraception must
exert an influence on the birth-rate.”

2.13 Write down in the form of the table of 2.1 the frequency groups
when (1) all .4*s are B's ; (2) all B*s are ^'s ; (3) all ^'s are B*s and all
B's area’s ; and the three similar tables when A and B are completely
disassociated.

2.14 Take the following figures for girls corresponding to those for boys
in Example 2.9, page 33, and discuss them similarly, but not necessarily
using exactly the same comparisons, to see whether the conclusion that
“ the connecting link between defects of body and mental dullness is the
coincident defect of brain which may be known by observation of abnormal
nerve signs ” seems to hold good.

At development defects ; S, nerve signs ; D, mental dullness.

10,000

(AB)

248

(A)

682

(AD)

307

(S)

850

(BD)

363

{D)

689

(ABD)

128

2.15 (Material from Census of England and Wales^ 1891, vol. 3.) The
following figures give the numbers of those suffering from single or com-
bined infirmities : (1) for all males ; (2) for males of 55 years of age and
over.

At blindness ; B, mental derangement ; C, deaf-mutism.

(1)

(2)

(1)

(2)

All Males

Males 55-

All Males

Males 55-

14,053,000

1,377,000

{AB)

183

(A)

12,281

5,538

(AC)

(B)

45,392

10,309

J^C)

299

(C)

7,707

746

(ABC)

Tabulate proportions per thousand, exhibiting the total association
between blindness and mental derangement, and the partial association
between the same two infirmities among deaf-mutes : (1) for males in
general ; (2) for those of 55 years of age and over. Give a short verbal
statement of the results.

ASSOCIATION OF ATTRIBUTES

2.16 (Material from same source as in Example 2.10).

The death-rate from cancer for occupied and retired males in general
(over 16) is 2-004 per thousand per annum, and for farmers 2*633.

The death-rates from cancer for occupied males under and over 45
respectively are 0-184 and 4-960 respectively. Of the farmers, 53-22
per cent are over 45.

Would you say that farmers were peculiarly liable to cancer ?

2.17 A population of males over 15 years of age consists of 7 per cent
over 65 years of age and 93 per cent under. The death-rates are 12 per
thousand per annum in the younger class and 110 in the older, or 18-86
in the whole population. The death-rate of males (over 15) engaged in
a certain industry is 26-7 per thousand.

If the industry be not unhealthy, what must be the approximate propor-
tion of those over 65 engaged in it (neglecting minor differences of age
distribution) ?

2.18 Show that if A and B are independent, while A and C, B and C are
associated, A and B must be disassociated either in the population of C's,
the population of y's, or both.

2.19 As 34 illustration of Exercise 2.18, show that if the following were
actual data, there would be a slight disassociation between the eye-colours
of husband and wife (father and mother) for the parents either of light-
eyed sons or not-light-eyed sons, or both, although there is a slight positive
association for parents at large.

A light eye-colour in husband, B in wife, C in son —

1,000

(AB)

358

{A)

622

(AC)

471

{B)

558

(BC)

419

(C)

617

2.20 Show that if {ABC)={a^y), {aBC)=(A^y), and so on (the case of
“ complete equality of contrary frequencies*' of Exercise 1.6, page 15),
A, B and C are completely independent if A and B, A and C, B and C
are independent pair and pair.

2.21 If, in the same case of complete equality of contraries,

{AB)-N/4===d^

{AC)^NI4^d,

(BC)-JV/4-53

show that

THTEORY OF STATISTICS

SO that the partial associations between A and B in the populations C and
7 are positive or negative according as

2.22 In the straight contests of a general election (contests in which one
Conservative opposed one Socialist and there were no other candidates)
66 per cent of the winning candidates (according to the returns) spent
more money than their opponents. Given that 63 per cent of the winners
were Conservatives, and that the Conservative expenditure exceeded the
Socialist in 80 per cent of the contests, find the percentages of elections
won by Conservatives (1) when they spent more and (2) when they spent
less than their opponents, and hence say whether you consider the above
figures evidence of the influence of expenditure on election results or no.
(Note that if the one candidate in a contest be a Conservative-winner-who
spends more than his opponent, the other must necessarily be a Socialist-
loser-who spends less — and so forth. Hence the case is one of complete
equality of contraries.)

2.23 Given that (A) IN=={B) /N=(C) and that (AB) IN={AC) jN

=7, find the major and minor limits to y that enable one to infer positive
association between B and C, i.e. (BC))N >

Draw a diagram on squared paper to illustrate your answer, taking x
and y as co-ordinates, and shading the limits within which y must lie in
order to permit of the above inference. Point out the peculiarities in the
case of inferring a positive association from two negative associations.

2.24 Discuss similarly the more complex case {A)IN==x, {B)IN=2x,

(0 IN^Sx—

(1) for inferring positive association between B and C given (AB) /N

=:{AC)IN=^y.

(2) for inferring positive association between A and C given (AB) /N

^{BC)IN=y,

(3) for inferring positive association between A and B given {AC) IN

^(BC)IN^y.

2.25 Draw a graph of the curve 7=2^ /(I +x^) for the range - 1 < < 1

and hence discuss the relationship between the coefficient of association Q
and the coefficient of colligation Y. Hence show, graphically or otherwise,
that the maximum difference between the two occurs when ^ is ±0* 786
approximately.

CHAPTER THREE

MANIFOLD CLASSIFICATION

Manifold classification

3.1 Instead of dividing the population under consideration into two parts
by a simple dichotomy, we may also divide it into a number of parts by
a similar process. For instance, we can extend the dichotomy of the
population of men into “ those with blue eyes '' and “ those not with blue
eyes to a threefold division : “ those with blue eyes those with
brown eyes,'' and “ those with neither blue nor brown eyes ", or into a
fourfold division by adding a fresh category, " those with grey eyes " ;
and so on.

Generally, our population may be divided first according to s heads,
• • • ^5 ; each of the classes so obtained into t heads, .

Bt ; each of these into u heads, C^, Cg, . . . Cu ; and so on.

This is called manifold classification,

3.2 The general theory of manifold classification for n attributes is
rather complicated, but its fundamental principles are very similar to
those which apply to dichotomy. A straightforward extension of the
methods of Chapter 1 will give the following results, which we are content
to announce without a formal proof —

{a) There are sxtxux ... ultimate classes.

(i) The total number of classes, including N and the ultimate classes,
is (s+l)(^+l)(w+l) • . •

{d) The data are completely specified by s x ^ X X ... algebraically
independent class-frequencies. Even if all these are not given, it may be
possible to set limits to the other class-frequencies.

For example, if the population of the United Kingdom is classified
geographically according to habitation in England, Wales, Scotland and
Northern Ireland ; by eye-colour into blue, brown, grey, green and the
remainder ; and by hair-colour into black, fair, red and the remainder ;
there will be 150 classes altogether, expressible in terms of 80 independent
class-frequencies.

3.3 Data so completely specified are very rare, and an elaborate discussion
of the general case would hardly be justified by its practical value. For

tHEORY OF STATISTICS

the remainder of this chapter, therefore, we shall be concerned solely
with the case of two characteristics, A and B,

Contingency tables

3.4 Let us suppose that the classification of the is s-fold and that
of the J5's is ^fold. Then there will be st classes of the type AmBn.

Generalising slightly the notation of previous chapters, let the frequency
of individuals Am be denoted by (Am) and of individuals AmBn by (AmBn).
The data can then be set out in the form of a table of t rows and s columns
as follows —

TABLE 3.1

Attribute

<^2

— As-1

Totals

Mi-Bi)

(A,B,} -

(A,B,}

(A,B,) -

(.As~iBj^)

(AsBJ

(B,)

(AiBt)

(A,Bt) -

— {As-tBt)

(AsBt)

(Bt)

Totals

(^i)

(A,) -

- (A..^

(A,)

In this table the frequency of the class AmBn is entered in the com-
partment common to the ^th column and the nth row ; the totals at the
ends of rows and at the feet of columns give the first order frequencies,
i.e. the numbers of Am’s and BnS ; and finally, the grand total in the
bottom right-hand comer gives the whole number of observations.

Such a table is called a contingency table. It is a generalised form
of the fourfold (2 X 2-fold) table in 2.1.

Example 3.1 — In Table 3.2 below the classification is 3 X 4-fold:
the eye-colours are classed under the three heads blue,'" ” grey or
green and brown,'' while the hair-colours are classed under four
heads, “ fair," “ brown," “ black " and red." Taking the first row,

TABLE 3.2 — Hah- and eye-colotirs of 6800 males in Baden

(Ammon, Zur Antkropologte der Bodmer)

Attribute

Fair

Haur-colour

Brown Black

Red

Total

Eye-colour

Blue.

1768

807

189

47'

2811

Grey or Green .

946

1387

746

3132

Brown

438

288

857

Total

2829

2632

1223

116

6800

MANIFOLD CLASSIFICATION

the table tells us that there were 2811 men with blue eyes noted, of whom
1768 had fair hair, 807 brown hair, 189 black hair and 47 red hair.
Similarly, from the first column, there were 2829 men with fair hair, of
whom 1768 had blue eyes, 946 grey or green eyes and 115 brown eyes.

Association in contingency tables

3.5 For the purpose of discussing the nature of the relation between
the ^’s and the any such table may be treated on the principles of
the preceding chapter by reducing it in different ways to a 2 x 2 -fold form*
It then becomes possible to trace the association between any one or more
of the ^’s and any one or more of the B's, either in the population at large
or in populations limited by the omission of one or more of the A’s, of the
B's, or of both.

If, for example, we desire to trace the association between a lack of
pigmentation in eyes and in hair, rows 1 and 2 may be pooled together as
representing the least pigmentation of the eyes, and columns 2 , 3 and 4
may be pooled together as representing hair with a more or less marked
degree of pigmentation. We then have —

Proportion of Hght-eyed with 1 3714 / 5943=46 per cent
fair hair . . . J

Proportion of brown-eyed with 1 j j ^ g

fair hair . , . J

The association is therefore well marked. For comparison we may trace
the corresponding association between the most marked degree of pigmen-
tation in eyes and hair, i.e. brown eyes and black hair. Here we must add
together rows 1 and 2 as before, and pool columns 1 , 2 and 4 — the column
for red being really misplaced, as red represents a comparatively slight
degree of pigmentation. The figures are —

Proportion of brown-eyed withl ggg /857 =34 per cent
black hair ’

Proportion of
black hair

The association is again positive and well marked, but the difference
between the two percentages is rather less than in the last case.

light-eyed withjggg / 5943 ==i 6

3.6 The mode of treatment adopted in the preceding two paragraphs
rests on first principles and, if fully carried out, gives us all the information
possible about the associations of the two attributes. At the same time
it is laborious if s and t are at all large. Moreover, in practical work we are
often concerned, not with the associations of individual A's with individual
B*s, but with finding the answer to a general question of the type : Are the
A*s on the whole distinctly dependent on the B's, and if so, is this depend-

THEORY OF STATISTICS

ence very close, or the reverse ? In fact, what we want is a coefficient
which will summarise the general nature of the dependence. We will
proceed to discuss two such coefficients.

Coefficients of contingency

3.7 If the ^'s and B's be completely independent in the population at
large, we must have for all values of m and n —

■ ■ • (3.1)

If, however, A and B are not completely independent, (AmBn) and {AmBn)o
will not be identical for all values of m and n. Let the difference be given

dmn~{AmBn) — [AmBn)Q . . . (^* 2 )

Let us note in passing the following properties of these quantities —

(1) In the first place, 8mn is not equal to dnm.

(2) In the second place, the S's are not all algebraically independent.
We have, in fact, for any particular m —

-(“5w 2 "f" • • » • • • A’^int

={AmBx) - +{A^B%) -

==(^„)_(^,;(Bi)+(S2)+ . . .

=0 (3.3)

A similar relation is true for any particular n.

Now there are st ^-quantities. In virtue of the relationship we have
just proved, for any particular m only (/ — I) of the ^quantities 8mn are
independent. Similarly, for any n only (s — 1) are independent. Hence
the total number of independent 8's> is (s — 1)(^— 1).

3.8 These ^-quantities indicate the extent of the associations, and we
expect a summarising coefficient to be built up from them in some way.
It would, however, be useless to add them together, for in virtue of the
relation of the preceding paragraph the sum is zero. We wish to construct
a coefficient which shall be independent of the signs of the (J-numbers.

We therefore define

and call the' square contingency.*'

MANIFOLD CLASSIFICATION

We then write —

. . . . (3.5)

arid call ^^jthe mean-square contingency/’

Clearly ')^ and being the sums of squares, cannot be negative. They
vanish if, and only if, every ^-number vanishes, in which case A and B
are independent.

Pearson’s coefficient of mean-square contingency

3.9 The quantity (j>^ is not quite suitable in itself to form a coefficient,
because its limits vary in different cases. Karl Pearson therefore proposed
the coefficient C, defined by

This is called the coefficient of mean-square contingency. In general,
no sign should be attached to the root, for the coefficient merely shows
whether two characters are or are not independent ; but in certain cases a
conventional sign may be used. Thus, in Table 3.2 slight pigmentation
of eyes and hair appear to go together, and the contingency may be
regarded as positive. If slight pigmentation of eyes had been associated
with marked pigmentation of hair, the contingency might -have been
regarded as negative.

3.10 The coefficient C has one serious disadvantage. Although, as
may be seen from its definition, it increases with (j>^ towards a limit 1, it
never reaches that limit. In fact, the maximum value which it can attain
depends on s and t, and reaches unity only for an infinite number of classes.
This may be briefly illustrated as follows. Replacing Smn in equation
(3.4) by its value in terms of {AmBn) and {AmBn)^, we have —

{AmBnY

(jimBnjQ

. (3.7)

and therefore, denoting the summation by 5,

.... (3.8)

Now suppose we have to deal with a fx^-fold classification in which
{Am)=^{Bm) for all values of m ; and suppose, further, that the association
between Am and Bm is perfect, so that {AmBm) ==^{Atn)=^{Btn) for all values
of m, the remaining frequencies of the second order being zero ; all the
frequency is then concentrated in the diagonal compartments of the table.

THEORY OF STATISTICS

and each contributes N to the summation S. The total value of S is
accordingly tN, and the value of C —

This is the greatest possible value of C for a symmetrical t x ^-fold classi-
fication, and therefore, in such a table, for —

t= 2,

C cannot exceed 0-707

„ 0-816

t= 4

„ 0-866

0-894

t= 6

„ 0-913

<= 7

„ 0-926

t= 8

„ 0-935

t= 9

0-943

t=io

„ 0-949

3.11 Hence, coeflficients calculated from different systems of classification
are not, strictly speaking, comparable. This is clearly undesirable. Two
coefficients calculated from the same data classified in two different group-
ings ought not to be very different.

It is as well, therefore, to restrict the use of the C-coefficient to 5 X 5 or
finer groupings. At the same time, the classification must not be made too
fine, or the value of the coefficient is largely affected by causal irregularities
arising from sampling fluctuations.^

Tschuprow’s coefficient

3.12 To remedy the defect to which we have just referred, Tschuprow
proposed the coefficient T, defined by

6 ^

= --7===^==^ . . . (3.9)

V(s-1)(<-1)

This coefficient varies between 0 and 1 in the desired manner when s==L
We have

C2=

and conversely.

. ( 3 . 10 )
. ( 3 . 11 )

^ Karl Pearson discussed a ** correction to be made to C calculated from coarsely
grouped data. The use of such corrections depends to some extent on assumptions
about the population, and may be regarded as attempts to bring the value of C closer
to a putative coefficient of correlation (cf. 10.20).

MANIFOLD CLASSIFICATION

Calculation of C and T

3.13 The calculation of C and T is simplified by the use of equation
(3.8), which enables us to replace the calculation of the by calcula-
tions based on frequencies of types {Am), {Bn) and (AmBn)^ All the<^e
quantities are contained in the contingency tables. The following example
will illustrate the method —

Example 3.2 — Consider the data of Table 3.2. (The classification is
only 3 X 4-fold and is therefore rather crude for calculating C, but it will
serve as an illustration of the form of the arithmetic.)

We require first of all the quantities {AmBn)^, i.e. the '‘independence'"
values. These are calculated directly from their definition

{AtnBi/^

{Am) (Bn)
N

and thus the value for the compartment in the wth column arid ^th row
is the product of the total frequencies in that column and row divided by
the whole frequency, e.g. (^jBi)q=2829 x 2811 /6800=1169, and so on.

It is convenient to tabulate the frequencies so obtained in a second
contingency table, as in Table 3.3.

TABLE 3.3 — Independence values of the frequencies for Table 3.2

Attribute

Hair-colour

Fair Brown Black Red

Eye-colour

Blue

Grey or Green ....
Brown . ...

1169 1088 506 48-0
1303 1212 563 53-4
357 332 154 14-6^

We now calculate the quantities

^ {AmBn)o

(1768)2/1169

2673-9

(946)2/1303

686-8

(115)2/357

37-0

(807)2/1088

598-6

(1387)2/1212

1587-3

(438)2/332

577-8

(189)2/506

70-6

(746)2/563

988-5

(288)2/154

538-6

f47)2/48-0

46-0

(53)2/53-4

52-6

(16)2/14-6

17-5

Total =S

=7875.2

56 THEORY OF STATISTICS

From equation (3.8)

IS~N_ /1075-2
“V S “V 7875-2

=V0- 1365 =0-37

and

0-1365

0-8635V6

^=V0-O645
= 0-25

The squares in such work may conveniently be taken from Barlow's
Tables of Squares, Cubes, etc,, or logarithms may be used throughout —
five-figure logarithms are quite sufficient.

It will be seen that T is less than C, This is not always true. Which-
ever coefficient we use, however, the contingency between pigmentation
of hair and eye is evident.

3.14 While such coefficients of contingency are a great convenience
in many forms of work, their use should not lead to a neglect of the more
detailed treatment of 3.5. Whether the coefficients be calculated or no,
every table should always be examined with care to see if it exhibits any
apparently significant peculiarities in the distribution of frequency, e.g.
in the associations subsisting between Am and Bn in limited populations.
A good deal of caution must be used in order not to be misled by casual
irregularities due to paucity of observations in some compartments of
the table, but important points that would otherwise be overlooked will
often be revealed by such a detailed examination.

3.15 Suppose, for example, that any four adjacent frequencies, say

(A m^n) tn+1 -^n)

are extracted from the general contingency table. If these are considered
as a table exhibiting the association between Am Sixid Bn in a population
limited to A^n alone, the association is positive, negative or

zero according as (A^B„} /(Atn+iBn) is greater than, less than, or equal
to the ratio (A^Bn+i) KAm+xByi+i)- The whole of the contingency table
can be analysed into a series of elementary groups of four frequencies like
the above, each one overlapping its neighbours, so that an s x ^fold table
contains ( 5 — l)(i— 1) such “ tetrads," and the associations in them all
can be very quickly determined by simply tabulating the ratios like

MANIFOLD CLASSIFICATION

(^mSn)/(^w+i-^n)i (^m^n+i) ctc., or perhaps better, the
proportions etc., for every pair of columns

or of rows, as may be most convenient. Taking the figures of Table 3.2
as an illustration, and working from the rows, the proportions run as
follows — •

For rows 1 and 2 For rows 2 and 3

1768/2714

0-651

946 /1061

0-892

807/2194

0-368

1387 /1825

0-760

189 /935

0-202

746 /1034

0-721

47 /lOO

0-470

53/69

0-768

In both cases the first three ratios form descending series, but the fourth
ratio is greater than the second. The signs of the associations in the six
tetrads are, accordingly,

+ + —

+ + ~

The negative sign in the two tetrads on the right is striking, the more so
as other tables for hair- and eye-colour, arranged in the same way, exhibit
just the same characteristic. But the peculiarity will be removed at once
if the fourth column be placed immediately after the first : if this be done,
i.e. if “ red " be placed between fair and brown instead of at the
end of the colour-series, the sign of the association in all the elementary
tetrads will be the same. The colours will then run fair, red, brown,
black, and this would seem to be the more natural order, considering the
depth of the pigmentation.

Isotropic contingency tables

3.16 A distribution of frequency of such a kind that the association
in every elementary tetrad is of the same sign, possesses several useful
and interesting properties, as shown in the following theorems. It will be
termed an isotropic distribution,

(1) In an isotropic distribution the sign of the association is the same not
only for every elementary tetrad of adjacent frequencies, hut for every set of
four frequencies in the compartments common to two rows and two columns,
e.g. {AmBn), {AmBn^-qj, {^Am-^pBn^-^*

For suppose that the sign of association in the elementary tetrads is
positive, so that

and similarly,

Then multiplying up and cancelling, we have —

[A fnBf^ {A n+1 ) ![> (^in+2'B«) n+l)

That is to say, the association is still positive though the two columns
and are no longer adjacent.

THEORY OF STATISTICS

(2) An isotropic distribution remains isotropic in whatever way it may
he condensed by grouping together adjacent rows or columns.

Thus from tlie first and third inequalities above we have, adding —

that is to say, the sign of the elementary association is unaffected by
throwing the (w^+l)th and (w-f'2)th columns into one.

(3) As the extreme case of the preceding theorem, we may suppose
both rows and columns grouped and regrouped until only a 2 x 2-fold
table is left ; we then have the theorem —

If an isotropic distribution be reduced to a fourfold distribution in any
way whatever by addition of adjacent rows and columns, the sign of the
association tn such fourfold table is the same as in the elementary tetrads of
the original table.

The case of complete independence is a special case of isotropy. For if
{An.Bn)=^{Am){Bn) /N

for all values of m and n, the association is evidently zero for every tetrad.
Therefore the distribution remains independent in whatever way the
table be grouped, or in whatever way the population be limited by the
omission of rows or columns. The expression complete independence "
is therefore justified.

From the work of the preceding section we may say that Table 3.2
is not isotropic as it stands, but may be regarded* as a disarrangement of
an isotropic distribution. It is best to rearrange such a table in isotropic
order, as otherwise different reductions to fourfold form may lead to
associations of different sign, though of course they need not necessarily
do so.

3.17 The following will serve as an illustration of a table that is not
isotropic and cannot be rendered isotropic by any rearrangement of the
order of rows and columns —

TABLE 3.4 — Showing the frequencies of different combinations of
eye-colours in father and son

1. Blue 2. Blue-green, grey 3. Dark grey, hazel 4. Brown
{Data of Galton, from Karl Pearson, Ph%L Trans., A, 1900, 195, 138 ; classification condensed.)

Son's

Eye-

colour

Father's Eye-colour

2 3

Total

194

335

124

284

137

109

244

Total

358

264

180

198

1000

MANIFOLD CLASSIFICATION 59

The following are the ratios of the frequency in column m to the sum
of the frequencies in columns m and m+l —

1 and 2

Columns

2 and 3

3 and 4

0-735

0-631

0-577

0-401

0-752

0-532

0-424

0-382

0-705

0-609

0-456

0-283

The order in which the ratios run is different for each pair of columns,
and it is accordingly impossible to make the table isotropic. The dis-
tribution of signs of association in the several tetrads is —

+ - +

- + -

- - +

The distribution is a curious one, the associations in tetrads round the
diagonal of the whole table being so markedly positive, and those in the
immediately adjacent tetrads equally markedly negative. Neglecting the
other signs, this is the effect that would be produced by taking an isotropic
distribution and then increasing the frequencies in the diagonal compart-
ments by a sufficient percentage. Comparison of the given table with
others from the same source shows that the peculiarity is common to the
great majority of the tables, and accordingly its origin demands explana-
tion. Were such a table treated by the method of the contingency
coefficient, or a similar summary method, alone, the peculiarity might not
be remarked.

Complete independence in contingency tables

3.18 It may be noted that in the case of complete independence the
distribution of frequency in every row is similar to the distribution in the
row of totals, and the distribution in every column similar to that in the
column of totals ; for in, say, the column the frequencies are given by
the relations —

and so on. This property is of special importance in the theory of variables.
Homogeneous and heterogeneous classification

3.19 The classifications both of this and of the preceding chapters
have one important characteristic in common, viz. that they are, so to
speak, homogeneous — the principle of division being the same for all

THEORY OF STATISTICS

the sub-classes of any one class. Thus A's and a*s are both subdivided
into B's and /?’s, A 2 ^, . . . As's into B^'s, . . . S/s, and

so on. Clearly this is necessary in order to render possible those coinpari-
sorib on which the discussions of associations and contingencies depend.
It we only know that amongst the .4’s there is a certain percentage of B's,
and amongst the a’s a certain percentage of C’s, there are no data for any
conclusion.

Many classihcations are, however, essentially of a heterogeneous
character, e.g. biological classifications into orders, general and species ;
the classifications of the causes of death in vital statistics and of occupa-
tions in the census. To take the last case as an illustration, the 1931
census of England and Wales divides occupations into 32 classes. Some
of these are not further subdivided — e.g. Fishermen 'h Others are sub-
divided into further general classes; e.g. Class 1 is divided into (1)
Employers, (2) Fumacemen, (3) Foundry Workers, (4) Smiths, (5) Metal
Machinists, (6) Fitters and (7) Other Workers. These sub-heads are
necessarily peculiar to the class under which they occur and their number
is arbitrary and variable, and different for each main heading ; but so long
as the classification remains purely heterogeneous, however complex it may
become, there is no opportunity for any discussion of causation within the
limits of the matter so derived. It h only when a homogeneous division
is^in some way introduced that we can begin to speak of associations and
contingencies,

3.20 This may be done in various ways according to the nature of
the case. Thus the relative frequencies of different botanical families,
genera or species may be discussed in connection with the topographical
characters of their habitats — desert, marsh or heath — and we may observe
statistical associations between given genera and situations of a given
topographical type. The causes of death may be classified according to sex,
or age, or occupation, and it then becomes possible to discuss the associa-
tion of a given cause of death with one or other of the two sexes, with a
given age-group or with a given occupation. Again, the classifications of
deaths and of occupations are repeated at successive intervals of time ; and
if they have remained strictly the same, it is also possible to discuss the
association of a given occupation or a given cause of death with the earlier
or later year of observation — ^i.e. to see whether the numbers of those
engaged in the given occupation or succumbing to the given cause of death
have increased or decreased. But in such circumstances the greatest
care must be taken to see that the necessary condition as to the identity of
the classifications at the two penods is fulfilled, and unfortunately it very
seldom^ is fulfilled. All practical schemes ot classification are subject to
alteration and improvement from time to time, and these alterations,
however desirable in themselves, render a certain number of comparisons
impossible. Even where a classification has remained verbally the same,
it is not necessarily really the same ; thus in the case of the causes of death,

MANIFOLD CLASSIFICATION

improved methods of diagnosis may transfer many deaths from one heading
to another without any change in the incidence of the disease, and so bring
about a virtual change in the classification. In any case, heterogeneous
classification should be regarded only as a partial process, incomplete untD
a homogeneous division is introduced either directly or indirectly, e.g, by
repetition.

Manifold classification as a series of dichotomies

3.21 From a theoretical point of view, manifold classification can be
regarded as compounded of a series ol dichotomies. Take, for example, a
case we have already considered, that of the classification of a population of
men according to the eye-colours blue, grey, brown and green. We could
have produced this fourfold division by three dichotomies. In fact,
dividing the population first into those with blue eyes and those with not-
blue eyes we get two classes. Then dividing again into those with brown
eyes and those with not-brown eyes we get four classes. This operation on
the class of blue-eyed men, however, results in one zero class, because there
are no men with blue eyes which are at the same time brown, and one class
which is, in fact, the class of blue-eyed men. Virtually, therefore, we have
three classes : those with blue eyes, those with browm eyes, and the re-
mainder. If we now dichotomise each of these into those with grey eyes
and those with not-grey eyes, we shall again get, neglecting the zero classes,
the four classes of the manifold classification.

3.22 It follows from this that any manifold classification can be regarded
as produced by a succession of divisions in which, at each stage, each
individual could fall into one of two alternatives, A or not-i4.

Put in another way, this means that the possible answers to an un-
ambiguous question can be reduced to a succession of answers of either
“ yes '' or '' no.'' For instance, suppose the question is, How old are you,
in years? " We can replace this question by the succession of questions,

Are you one year old ? Are you two years old Are you

120 years old ? " An answer of ‘‘ 47 " to the first-mentioned question can
then be expressed as an answer of “ No " to the first 46 of these questions,

Yes " to the 47th and No " to the rest.

Similarly, an answer to the question, “ What is your name ? " can be
reduced to the questions, '' Is the first letter of your name A ? " *‘ Is the
first letter B Is the second letter A ? " and so on. Replies to

a more general question can be reduced to the same form by a convenient
classification ; e.g. the replies to the question, “ Are you in favour of war?”
can be classified in the four forms “ Favourable without qualification,”

Favourable with some qualification." Unfavourable without qualifica-
tion," Unfavourable with some qualification," and the answers to the
questions can be reduced to answers ‘ yes " or no " to the questions, "Are
you, without qualification, in favour of war ? " and so on.

THEORY OF STATISTICS

Recording classified information on punched cards

3.23 The information about an individual, considered as a member
of a population, is information whether he does or does not fall into the
alternative classes which, as we have just seen, compose the most general
homogeneous classification of the population. If we imagine each indi-
vidual filling in a questionnaire about himself, the totality of answers may,
by suitably expressing the questions, be expressed as a number of yes's ”
and “ no’s," and these replies express all the information about the
individual.

This simple fact allows us to record the data in a most convenient way.
Each individual is allotted a card, which is divided into a number of cells.
Each cell corresponds to one of the dichotomies or simple questions the
answers to which constitute the information. If the answer is Yes,” a
hole is punched in the cell ; if the answer is "No," the cell is left un-
touched.

The card of any individual will thus be like a complicated bus ticket,
with holes punched in various places. The punching is usually performed
either by hand with a ticket collector's punch, or with a machine similar
in principle to the typewriter. The totality of punched cards forms a
miniature of our population — each individual has a card on which is
recorded the whole of the information about him.

The use of this system lies in the fact that punched cards are easily
handled and sorted by machinery. If, for example, we want to know a
particular class-frequency, we can adjust certain electrical, pneumatic or
mechanical stops, and the machine will segregate all the cards in the class
and count them for us.

3.24 A similar device has been applied to the sorting of data by hand.
A card is prepared with a row of circular holes punched all the way round
near its edge but so that no hole is open to the edge. Each hole corre-
sponds to a dichotomy or a simple question. When preparing the card, if
the indmdual falls into the A class, or the answer to the question is " Yes,"
a piece is clipped out of the card so that the hole is now open to the edge.
If the individual falls into the not-4 class, or the answer to the question is
" No,” the hole is left alone.

To separate the ^'s from the not-^'s, or the " yes " cards from the
" no " cards, they are arranged in a vertical plane so that corresponding
cells are similarly placed. A skewer is then inserted in the appropriate
hole and lifted. The not-A cards are lifted out, whilst the A cards fall
away, since the piece of card between the hole and the edge has been cut
away. By repeating the operation with the skewer in the appropriate
holes we can isolate the cards in any given class. These can then be
counted and the size of the class-frequency determined.

3.25 The labour of punching cards and the expense of machinery is
justified only when the number of individuals is large and the number of

MANIFOLD CLASSIFICATION 63

ultimate classes is also large. This arises, for example, in the taking of
a census of population.

Numerically defined attributes

3.26 The attributes we have instanced in the foregoing pages have
usually been of a qualitative kind. The methods described are, however,
applicable to data classified on a numerical basis. Consider, for example,
the following table —

TABLE 3.5 — Families deficient in room space
Their number in 95 crowded London wards
(Census of 1931, Housing Report, p xxxu)

Families

deficient

Standard room requirement
(rooms)

2 3 4 5 6 7 8

Totals

1 room

12,999 18,198 7,724 2,170 164 19

41,274

2 rooxns

3,054 4,479 1,448 221 15 1

9,218

3 rooms

310 508 106 4 1

929

4 rooms

10 21 4

Totals

12,999 21,252 12,513 4,136 512 42 2

51,456

The distinction between successive rows and columns is not quite of the
kind of Table 3.2. In the latter, for instance, we drew a line between black
hair and brown, a line which could be drawn by anybody who was not
colour-blind, although there may be border-line cases of mixed colours
which would present difficulty. But in Table 3.5 above the line is drawm
by counting — a much more precise operation. Moreover, the rows and
columns have a certain natural order given by the numerical sequence.
It would seem absurd to put the column which is headed “ two rooms
between those headed three rooms ” and ” four rooms,*' but in Table 3.2
there is no a priori reason for putting black '* between brown " and
red."

3.27 We might also have a contingency table in which the attributes
were measurable quantities, and the rows and columns of the table de-
termined by ranges of those quantities. This, again, is slightly different
from the case of the previous paragraph, for these ranges are to a large
extent arbitrary, whereas in Table 3.5 the indivisible nature of the room
compels us to count in units of at least one room.

3.28 Finally, we may have a table which is given by one qualitative
attribute and one quantitative attribute. Consider, for example, the
following —

THEORY OF STATISTICS

TABLE 3.6 — Weight and mentality in a selection of criminals

{Data from M, H. Whiting, “ On the Association, of Temperature, Pulse and Respiration with Physique and
Intelligence m Criminals,” Bwmetrika^ 1912, U, 1)

Mentality

90-120

Weight (lb)

120-130 130-140 140-150

150

upward

Totals

Normal

106

124

396

Weak

Totals

128

121

139

493

3.29 The methods of the previous chapters are applicable also to such
tables. Numerically measurable quantities may, however, be treated by
other methods, to which we shall come in due course. We mention the
point here in order to remove any possible idea that the theory of attributes
is concerned solely with qualitative classification, and is not appropriate
to the more precise data given by a numerically assessable attribute.

SUMMARY

1. The division of a population according to an attributed into a number
of heads is called manifold classification. This is an extension of the idea
of dichotomy, in which the population is divided into two parts only.

2. Manifold classification according to two attributes A and B gives
rise to a contingency table.

3. Association in a contingency table may be examined by reducing it
in a number of ways to a 2 x2 table.

4. We define

— (d tn’^n) 0

The " square contingency '' is given by —

The " mean-square contingency ** by —

MANIFOLD CLASSIFICATION 65

5, Pearson’s “ coefficient of mean-square contingency " is defined by —

6. Tschuprow’s '' coefficient of contingency ” is defined by —

7. Certain types of table, known as isotropic contingency tables, possess
special features of some importance.

8. Any manifold classification may be regarded as a succession of
dichotomies. This fact is the basis of the use of punched cards for record-
ing and analysing statistical data.

9. Manifold classification may arise not only from an attribute which
is specified under heads of a qualitative kind, but also from a quantitative
attribute specified by counting or measurement.

EXERCISES

3.1 (Data from Karl Pearson, ** On the Inheritance of the Mental and
Moral Characters in Man,” Jour, of the Anthrop, Inst,, vol. 33, and
Btometrika, vol. 3.) Find the coefficient of contingency (coefficient of
mean-square contingency) for the two tables below, showing the resem-
blance between brothers for athletic capacity and between sisters for
temper. Show that neither table is even remotely isotropic. (As stated
in 3.11, the coefficient of contingency should not as a xuh be used for
tables smaller than 5 x 5-fold : these small tables are given to illustrate
the method, while avoiding lengthy arithmetic.)

A. Athletic capacity

Second Brother

Athletic

First Brother

Betwixt

Non-

athletic

Total

Athletic .

906

140

1066

Betwixt .

105

Non-athletic

140

370

519

Total

1066

105

519

1690

THEORY OF STATISTICS

B. Temper

Second Sister

Quick

Good-natured .

SuUen

First Sister

Good-

Quick natured Sullen

Total

198«(U 177 77

177aRr 996 ^3'*'^ 16512.
77 ”}]p 165 120^ j

^ 452

r 1338

362

Total

452 1338 362

2152

3.2 Calculate T and C for the foUowing table, and trace the association
between the progress of building and the urban character of the district —

Houses in England and Wales

{Census ef 1901. Summary Table X, 000‘s omitted)

Inhabited

Unin-

habited

Building

Total

Adm. County of London .

571

616

Other urban distncts

4064

285

4394

Rural districts

1625

124

1761

Total for England and Wales

6260

449

6771

3.3 Show that for a given s and t, C and T are equal for two values of
one of which is zero ; that for between these values C > T \ and

that for greater than the higher value T > C.

3.4 Find whether the following contingency table is isotropic, and if it
is not, ascertain whether it can be arranged in an isotropic form —

Totals

193

235

467

300

103

576

Totals

625

234

115

158

104

1236

MANIFOLD CLASSIFICATION

3.5 Calculate C and T for the table of the previous example.

3.6 Show that in a positively isotropic contingency table,

ii — ^ 11 — and IS > —

1,000 subjects of English, French, German, Italian and Spanish
nationality were asked to name their preferences among the music of those
five nationalities. The results were as follows (l=English, 2=French,
3=Gennan, 4=Italian, 5=Spanish) —

Nationality

subject

Nationality of music preferred

2 3 4

Totals

200

107

200

^ i

200

Totals

179

298

243 202

1000

Discuss the association between the nationality of the subject and the
nationality of the music preferred.

3.8 In Table 3.6 calculate C and J, and discuss the light thrown by this
table on the association between physique and intelligence in the criminals
of the data.

3.9 Show that for a 2x2 contingency table in which the frequencies are

2 -- [ad—'bc)^

and hence find C and T in terms of a, 6, c, d.

3.10 In a paper discussing whether laterality of hand is associated
with laterality of eye (measured by astigmatism, acuity of vision,

THEORY OF STATISTICS

etc.) T. L, Woo obtained the following results {Biometrika, vol. 20A,
pp. 79-148)—

Manual laterality
as determined
by a balancing
test

Ocular laterality for general astigmatism

'* Left-eyed ” Ambiocular “ Right-eyed

Totals

Left-handed

124

Ambidextrous .

Right-handed .

105

214

Totals

118

195

100

413

Show that laterality of eye is only slightly associated with laterality of
hand.

CHAPTER FOUR

FREQUENCY-DISTRIBUTIONS

Variables

4.1 As we emphasised at the close of the last chapter, the methods
of the theory of attributes are applicable to all observations, whether
qualitative or quantitative. We have now to proceed to the consideration
of special processes adapted to the treatment of quantitative data, but
not as a rule available for the discussion of purely qualitative observations
(though there are some important exceptions to this statement, as suggested
in 1.2).

A measurable quantity which can vary from one individual to another
is called a variable^ and this section of our work may be termed the theory
of variables.

As common examples of variables which are subject to statistical
treatment we may cite birth- and death-rates, prices, wages, barometer
readings, rainfall records, and measurements or enumerations (e.g. of
glands, spines or petals) on animals or plants.

Quantities which can take any numerical value within a certain range
are called continuous variables. Such, for example, are birth-rates and
barometric readings. Quantities which can take only discrete values
are called discontinuous variables. This class, for instance, would include
data of the number of petals on flowers or the number of rooms in a house.

Frequency-distributions

4.2 If some hundreds or thousands of values of a variable have been
noted merely in the arbitrary order in which they occur, the mind cannot
properly grasp the significance of the record. We must condense the
data by some method of ranking or classification before their characteristics
can be comprehended.

One way of doing this would be to dichotomise the data by classifying
the individuals as A's or not-il's, according as the value of the variable
exceeded or fell short of some given value. But this is too crude, and
the sacrifice of information is too great. A manifold classification,
however, avoids the crudity of the dichotomous form, since the classes
may be made as numerous as we please. Moreover, numerical measure-
ments lend themselves with peculiar readiness to a manifold classification,

1 It is also called a variate. We shall use the two terms as synonymous.

THEORY OF STATISTICS

for the class limits can be conveniently and precisely defined by assigned
values of the variable.

4.3 For convenience, the values of the variable chosen to define the
successive classes should be equidistant, so that the numbers of observa-
tions in different classes are comparable.

The interval chosen for classifying is called the class-interval, and the
frequency in a particular class-interval is called a class-frequency.

Thus, for measurements of stature, the class-interval might be 1 inch,
or 2 centimetres, and the class-frequencies would be the numbers of indi-
viduals whose statures fell within each successive inch or each successive
2 centimetres of the scale ; returns of birth- or death-rates might be
grouped to the nearest unit per thousand of the population ; returns of
wages might be classified to the nearest shilling, or, if it is desired to obtain
a more condensed table, to the nearest five or ten shillings. Discon-
tinuous variables to a great extent determine their own class-intervals,
which must either be equal in width to the unit amount of variation, or
equal to some multiple of it. For example, in enumerations of the
number of rooms in a house we naturally take our class-interval to be
one room ; in enumerations of the petals on a flower we may take one
petal or, if the range of variation is very great, say five petals or more.

4.4 The manner in which the class-frequencies are distributed over
the class-intervals is spoken of as the frequency-disirihuiion of the variable.

A few illustrations will make clearer the nature of such frequency-
distributions, and the service which they render in summarising a long
and complex record.

TABLE 4.1 — Showing the number of local government areas in England with specified
birth-rates per thousand of population

(Matenal from the Registrar-Gerieral's Statistical Review of England and Wales for 1933)

Number of districts

Birth-rate

with birth-rate

Birth-rate

with birth-rate

between

Himts stated

limits stated

1-5- 2-5

13*5-14-5

271

2-5- 3*5

14*5-15-5

190

3-5- 4*5

15-5-16-5

127

4*5- 5*5

16*5-17-5

S*5~ 6*5

17*5-18-5

6-5- 7-5

18*5-19-5

7'5- 8-5

19*5-20-5

8-5- 9-5

20-5-21-5

9^5-10*5

21-5-22-5

' 4

10*5^11 *5

131

22*5-23-5

11 •5-12-5
12*5-13-5

192

242 '

23-5-24-5

Total

1567

FREQUENCY-DISTRIBUTIONS ^X

{a) Table 4.1. In this illustration the birth-rates per thousand of
the population in 1933 of 1,567 local government areas of England have
been classified to the nearest unit ; i.e. the number of districts has been
counted in which the birth-rate was between 1*5 per thousand and 2*5,
between 2*5 and 3*5, and so on. The frequency-distribution is shown by
the table.

Although a glance through the original returns, which are spread amongst
many other figures over 42 pages, fails to convey any definite impression,
a brief inspection of the above table brings out a number of important
points. Thus, we see that the birth-rates range, in round numbers, from
2 to 24 per thousand ; that the birth-rates in some 75 per cent of the
districts lie within the narrow limits 10-5 to 16 • 5, the rates most frequent
being near 14 ; and so on. It may be remarked that some of the areas
are very small, with no more than 10 or 20 births, and these account
mainly for the extremely divergent rates.

(6) Table 4.2. The numbers of stigmatic rays on a number of Shirley
poppies were counted. As the range of variation is not great, the unit
is taken as the class-interval. The frequency-distribution is given by
the following table —

TABLE 4.2 — Showing the frequencies of seed capsules on certain Shirley poppies with
different numbers of stigmatic rays

(Cited from G. Udny Yule, Bvmetnka, 1902, 2, 89)

Number of
stigmatic
rays

Number of
capsules
with said
number of
stigmatic rays

Number of
stigmatic
rays

Number of
capsules
with said
number of
stigmatic rays

*14

302

234

128

106

152

238

305

315

Total

1905

The numbers of rays range from 6 to 20, the most usual numbers being
12, 13 or 14.

(c) Table 4.3. 206 screws were taken as they came off the lathe which
was turning them. Their lengths, which should have been 1 inch, were
measured. The following table shows the screws classified by the number

THEORY OF STATISTICS

of thousandths of an inch by which they exceeded or fell short of 1 inch
in length —

TABLE 4.3 — Showing the frequencies of screws classified according to the extent to
which they varied in length from the standard of 1 inch

Difference in length
from 1 mch
(Thousandths of an
inch)

Number of
screws

Difference in length
from 1 inch
(Thousandths of an
inch)

Number of
screws

-6 to -5

-f“ 1 to -j- 2

-5 to -4

-j- 2 to -}- 3

—4 to —3

-}- 3 to -f" 4

-3 to -2

-j- 4 to -f- 5

-2 to -1

-j- 5 to -j- 6

— “ 1 to 0

0 to -fl

Total

206

It will be seen that the maximum frequency, i.e. 34, occurs for screws
from 0-001 to 0-002 inch in excess of the standard. About 80 per cent
lie in the range three-thousandths of an inch on either side of the standard.

4.5 Expanding slightly the brief description we have given, tables
setting out frequency-distributions are formed in the following way —

(1) The magnitude of the class-interval is first fixed. In Tables 4.1,
4.2 and 4.3 one unit was chosen.

(2) The position or origin of the intervals must then be determined ;
e.g. in Table 4.1 we must decide whether to take as intervals 9-10, 10-11,
11-12, etc., or 9-5-10-5, 10-5-11-5, 11-5-12-5, etc.

(3) This choice having been made, the complete scale of intervals is
fixed and the observations are classified accordingly.

(4) The process of classification being finished, a table is drawn up on
the general lines of Tables 4. 1-4.3, showing the total number of observa-
tions in each class-interval.

It is necessary to make a few remarks about each of these heads.

Magnitude of class-interval

4.6 As already remarked, in cases where the variation proceeds by
discrete steps of considerable magnitude as compared with the range of
variation, there is very little choice as regards the magnitude of the class-
interval. The unit will in general have to serve. But if the variation
be continuous, or at least takes place by discrete steps which are- small
in comparison with the whole range of variation, there is no such natural
class-interval, and its choice is a matter for judgment.

The two conditions which guide the choice are these : {a) We desire
to be able to treat aU the values assigned to any one class, without serious

FREQUENCY-DISTRIBUTIONS

error, as if they were equal to the mid-value of the class-interval, e.g.
as if the birth-rate of every district in the first class of Table 4.1 were
exactly 2-0, the birth-rate of every district in the second class 3*0, and
so on ; (6) for convenience and brevity we desire to make the interval
as large as possible, subject to the first condition. These conditions will
generally be fulfilled if the interval be so chosen that the whole number
of classes Hes between 15 and 25. A number of classes less than, say,
ten leads in general to very appreciable inaccuracy, and a number over,
say, thirty makes a somewhat unwieldy table. A preliminary inspection
of the record should accordingly be made and the highest and lowest
values be picked out. Dividing the difference between these by, say,
twenty-five, we have an approximate value for the interval. The actual
value should be the nearest integer or simple fraction.

Position of intervals

4.7 The position or starting-point of the intervals is, as a rule, more or
less a matter of indifference. It can therefore be chosen as is most
convenient for the particular case under discussion, e.g. so that the limits
of the intervals are integers, or, as in Table 4.1, so that the mid- values are
inte’gers. It may also be chosen so that no limits correspond exactly
to any recorded value of the variate, in order to obviate any difficulty
in deciding to which class a particular individual should be assigned
(cf. 4.9).

The location of the intervals is, however, important when the values
of the variate tend for some reason to cluster round particular values.
Such a case arises, for instance, in age returns, owing to the tendency
to state a round number where the true age is unknown, or a reluctance
to admit one’s real age.^ It is also common wherever there is some
doubt as to the final digit in reading a scale, and scope is given to the
idiosyncrasies of the observer.

Table 4.4 shows results for four observers as illustrations, the frequencies
being reduced for comparability to a total of 1 ,000. Column A is based
on measures by G. U. Yule, on drawings, to the nearest tenth of a milli-
metre. It is recognised, of course, that measures cannot really be made to
such a degree of precision ; but the measurer believed that he was making
them carefully, and as they were made with a Zeiss scale, in which the
divisions are ruled on the under side of a piece of plate-glass, readings
were unaffected by parallax. Nevertheless, it will be seen that the
zeros, and also 2, 8 and 9, were heavily over-emphasised — an odd selection
of preferences! On the whole, the centre of the millimetre was neglected
and measures piled up at the two ends.

The data for columns B, C and D are all drawn from the same published
report, and refer to sundry head measurements taken on the living subject.

^This effect is practically the same for men as for women. Cf. Table I in the Appen-
dix to the paper cited in the heading to Table 4.4 above.

THEORY OF STATISTICS

On the basis of a statement in the introduction to the report, it was possible
to compile the data separately for the three assistants (B, C, D) who had
done the actual measuring. It will be seen that B was rather good : there
is a relatively slight excess at 0 and 5, but otherwise his measurements are
fairly uniformly distributed. C was decidedly not good, rounding off nearly
one measurement in two to the nearest centimetre or half-centimetre. D
was simply outrageously bad — so bad that it might have been better not
to publish his measurements. Nearly 57 per cent of his measurements
were made only to the nearest centimetre or half-centimetre — a quite
inadequate degree of precision for head measurements often only a few
centimetres in magnitude.

TABLE 4.4 — Frequency-distributions of final digits in measurements by four observers

(G U. Yule, “ Oa Reading a Scale,” J, Rov Stai, Soc , 1927, 90, 570)

Final digit

Frequency of final digit per 1,000 for observer

158

122

251

358

125

100

112

222

211

126

101

129

Total

1001

999

1000

Actual ob“\
servations j

1258

3000

1000

When there is any possibility of clustering of variate values it is as
well to subject the data to a close examination before finally filing on
the method of classification. On the whole, the intervals should be
arranged as far as possible so that the values round which the clustering
occurs fall towards the interval mid-values. This procedure avoids
sensible error in the assumption that the interval mid-value is approxi-
mately representative of the values of the class.

Classification

4.8 The scale of intervals having been fixed, the observations may
be classified. If the number of observations is not large, it will be sufficient
to mark the limits of successive intervals in a column down the left-hand
side of a sheet of paper, and transfer the entries of the original record
to this sheet by marking a 1 on the line corresponding to any class for
each entry assigned thereto. It saves time in subsequent totalling if

FREQUENCY-DISTRIBUTIONS 75

each fifth entry in a class is marked by a diagonal across the preceding
four, or by leaving a space.

The disadvantage in this process is that it offers no facilities for checking :
if a repetition of the classification leads to a different result, there is no
means of tracing the error. If the number of observations is at all con-
siderable and accuracy is essential, it is accordingly better to enter the
values observed on cards, one to each observation. These are then
dealt out into packs according to their classes and the whole work checked
by running through the pack corresponding to each class, and venfying
that no cards have been wrongly sorted.

4.9 In some cases difficulties may arise in classif 3 dng, owing to the
occurrence of observed values corresponding to class-limits. Thus, in
compiling Table 4.1 some districts will have been noted with birth-rates
entered in the Registrar-General’s returns as 16*5, 17*5 or 18-5, any one
of which might at first sight have been apparently assigned indifferently
to either of two adjacent classes. In such a case, however, where the
original figures for numbers of births and population are available, the
difficulty may be readily surmounted by working out the rate to another
place of decimals; if the rate stated to be 16*5 proves to be 16*502, it
will be sorted to the class 16*5-17*5 ; if 16*498, to the class 15*5-16*5.
Birth-rates that work out to half-units exactly do not occur in this example,
and so there is no real difficulty.

In the case of Table 4.3, again, there is little difficulty in knowing the
class to which an individual should be assigned.

Difficulties of this type may, in fact, alwaj^s be avoided if they are
borne in mind in fixing the class-intervals, by fixing the intervals to a
further place of decimals or a smaller fraction than the values in the
original record. Thus, if statures are measured to the nearest centimetre,
the class-intervals may be taken as 150 • 5-151 • 5, 1 51 * 5-152 * 5, etc.; if to the
nearest eighth of an inch, the intervals may be 59^-60^1-, 60-J|-61||,
and so on.

If the difficulty is not evaded in any of these ways, it is usual to assign
one-half of an intermediate observation to each adjacent class, with the
result that half-units occur in the class-frequencies (cf. Table 4.9, p. 86).
The procedure is rough, but probably good enough for practical purposes ;
strict precision is usually unattainable, for in point of fact the odd way in
which different individuals read a scale, for example, renders it impossible
to assign exact limits to intervals.

Tabulation

4.10 As regards the actual drafting of the final table there is little
to be said, except that care should be taken to express the class-limits
clearly and, if necessary, to say how the difficulty of intermediate values
has been met or evaded. The class-limits are perhaps best given as in

THEORY OF STATISTICS

Tables 4.1 and 4.3, but may be more briefly indicated by the mid-values of
the class-intervals. Thus, Table 4. 1 might have been given in the form —

Birth-rate per 1,000 to
the nearest unit

etc.

Number of districts with
said birth-rate

etc.

It is also permissible to write the table in the form —

Interval

1*5-

2*5--

3-5-

etc.

Frequency

etc.

it being understood that the closing point of any interval is the starting
point of the following interval. Cf. Table 4.11 below.

It should be noticed that the method of defining class-intervals adopted
in Table 4.3 leaves the class-limits uncertain unless the degree of accuracy
of the measurements is also given. Thus, in a table giving frequencies of
men in certain height-ranges of 1 inch in width, say “ 57 and less than 58/'
etc., if measurements were taken to the nearest eighth of an inch, the class-
limits are really 56||-57-J|, 57^~58j^, etc.; if they were only taken to
the nearest quarter of an inch, the limits are 56-|-57|, 57|-58f, etc. With
such a form of tabulation a statement as to the number of significant figures
in the original record is therefore essential. It is better, perhaps, to state
the true class-limits and avoid ambiguity.

4.11 The rule that class-intervals should be all equal is one that is
very frequently broken in official statistical publications, principally in
order to condense an otherwise unwieldy table, thus not only savdng space
in planting but also considerable expense in compilation, or possibly, in the
case of confidential figures, to avoid giving a class which would contain
only one or two observations, the identity of which might be guessed. It
would hardly be legitimate, for example, to give a return of incomes relating
to a limited district in such a form that the income of the two or three
wealthiest men in the district would be clear to any intelligent reader with
local knowledge.

If the class-intervals be made unequal, the application of many statis-
tical methods is rendered awkward, or even impossible. Further> the
relative values of the frequencies are misleading, so that the table is not
perspicuous. Thus, consider the first two columns of Table 4.5, showing
the number of persons liable to sur-tax and super-tax classified according
to their annual income. On running the eye down the column headed

Number of Persons/' the attention is at once caught by the three irregu-

FREgUENCY-DISTRIBUTIONS

larities at the classes “ £3fi00 and not exceeding ;^^4,000/' " £8,000 and
not exceeding £10,000/’ and “ £10,000 and not exceeding £15,000.” But
these have no real significance ; they are merely due to changes in the
magnitude of the class-interval at those points. A further change occurs
at the £30,000 and at the £50,000 mark, although the attention is not
directed thereto by an}^ marked irregularity in the frequencies.

TABLE 4.5 — The numbers of persons in the United Kingdom liable to sur-tax and
super-tax in the year beginning 5th April 1931

Classified according to the magnitudes of their annual incomes
(From the Statistical Abstract for the Uaited Kingdom for the Years 1913 and 1919-32, Cmd 4489)

Annual income
{£000)

Number of
persons

Frequency per
£500 interval

2 and not exceeding 2*5

23,988

2 5 „

15,781

3 ,,

17,979

8,989

4 ,, ,,

9,755

4,877

5,921

2,960

3 ,, ,,

3,729

1,864

2,546

1,273

S 1 , 1 ,

3,193

798

10 „

3,616

362

1,328

I 133

20 „

679

25 „

378

30 , , , ,

372

192

50 „

182

75 „

100

100 and over

Total number of persons

89,790

—

To make the class-frequencies really comparable inter se they must first
be reduced to a common interval as basis, say £500, by dividing the third
and subsequent numbers by 2, the eighth by 4, and so on This gives
the mean frequencies tabulated in the third column of Table 4.5. The
reduction is, however, impossible in the case of the last class, for we are
told only the number of persons with an income or£ 1 00,000 and upwards.
Such an indefinite class is in many respects a great inconvenience, and
should always be avoided in work not subjected to the necessary limitations
of official publications.

4.12 The general rule that intervals should be equal must not be held
to bar the analysis by smaller equal intervals of some portion of the range
over which the frequency varies very rapidly. In Table 4.11, page 89,
for example, giving the numbers of deaths from scarlet fever at successive
ages, it is desirable to give the numbers of deaths in each year for the first
five years, so as to bring out the rapid rise to the maximum in the third
year of life.

THEORY OF STATISTICS

Graphical representation : frequency-polygon and histogram
4.13 It is often convenient to represent the frequency-distribution
by means of a diagram which conveys to the eye the general run of the
observations. The following short table, giving the distribution of head-
breadths for 1,000 men, will serve as an example —

TABLE 4.6 — Showing the frequency-distribution of head-breadths for students at

Cambridge

Measurements taken to the nearest tenth of an inch
(Cited from W R. Macdonell, Bionietnka, 1902, 1, 220)

Head-breadth
in inches

Number of
men with said
head-breadth

Head-breadth
m inches

Number of
men with said
head -breadth

5-5

6 3

5-6

6 4

5-7

6*5

5*8

6 6

5-9

131

6*7

6-0

236

6*8

6*1

185

6*2

142

Total

1000

Taking a piece of squared paper ruled, say, in inches and tenths, mark
off along a horizontal base-hne a scale representing class-intervals ; a
half-inch to the class-interval would be suitable. Then choose a vertical
scale for the class-frequencies, say 50 observations per interval to the inch,
and mark off, on the verticals or ordinates through the points marked 5*5,
5*6, 5 • 7, . . . at the centres of the class-intervals on the base-line, heights
representing on this scale the class-frequencies 3, 12, 43, . . . The diagram
may then be completed in one of two ways : (1) as a frequency-polygon,
by joining up the marks on the verticals by straight lines, the last points at
each end being joined down to the base at the centre of the next class-
interval (fig. 4.1) ; or (2) as a column diagram or histogram, short
horizontals being drawn through the marks on the verticals (fig. 4.2), which
now form the central axes of a series of rectangles representing the class-
frequencies.

4.14 The student should note that in any such diagram, of either form,
a certain area represents a given number of observations. On the scales
suggested, 1 inch on the horizontal represents 2 intervals, and 1 inch
on the vertical represents 50 observations per interval : 1 square inch
therefore represents 50x2=100 observations. The diagrams are, how-
ever, conventional : in both cases the whole area of the figure is pro-
portional to the total number of observations, but the area over every
interval is not correct in the case of the frequency-polygon, and the
frequency of every fraction of any interval is not the same, as suggested
by the histogram. The area shown by the frequency-polygon over any

FREQUENCY-DISTRIBUTIONS

interval with an ordinate 3/3 (fig. 4.3) is only correct if the tops of the three
successive ordinates y-i, y%, yz lie on a line, i.e. if y^ =\{y\ +>' 3 ). the areas of
the two little triangles shaded in the figure being equal. If y^ fall short of
this value, the area shown by the polygon is too great ; if exceed it,

THEORY OF STATISTICS

the area shown by the polygon is too small ; and if, for this reason, the
frequency-polygon tends to become very misleading at any part of the
range, it is better to use the histogram.

4.15 The histogram may also be used when the class-intervals are
unequal. The construction of the previous section is easily adapted to
such cases. All that is necessary is to describe an area equal, on the scale
adopted, to the frequency in a particular interval ; this is done, as before,
by erecting at the centre of the interval an ordinate equal in length to
the total frequency divided by the width of the interval.

An example of this kind of con-
struction is given in fig. 4.11 (Table
4.11). The frequencies of deaths for
ages over 5 years are given in 5-y early
periods, whereas those for ages under
5 years are given in 1 -yearly periods.
On the scale indicated, therefore, the
height of the cell of the histogram cor-
responding to the ages 2-3 years is
89, the class-frequency ; that of the
cell corresponding to the ages 5-10 is
42*6, i.e. 213 divided by 5. Hence the
Fig. 4.3 areas of the two cells are, to the scale

adopted, 89 and 213, respectively, so that the areas accurately represent
the frequencies.

Frequency-curves

4.16 If the class-intervals be made smaller, and at the same time the
number of observations increased so that the class-frequencies may
remain finite, the polygon and the histogram will approach more and

Fig. 4.4

FREQUENCY-DISTRIBUTIONS 8l

more closely^ to a smooth curve. Such an ideal limit to the polygon or
the histogram is called a frequency-curve. It is a concept of supreme
importance in statistical theory.

In the frequency-curve the area between any two ordinates wjiatever
is proportional to the number of observations falling between the corre-
sponding values of the variable. Thus, the number of observations
falling between the values of the variable and in fig. 4.4 will be
proportional to the area of the shaded strip in the figure ; the number of
observed values greater than will be given by the area of the curve to
the right of the ordinate at ; and so on.

4.17 When we come to consider the theory of sampling we shall regard
the frequency curve as representing a population from which the actual
data are a specimen. The frequency-polygon and the histogram will then
be approximations to the curve, but will diverge from it to some extent
owing to fluctuations of sampling. For the present we must defer a closer
inquiry into this subject. We may remark, however, that when the
number of observations is considerable — say a thousand at least — the
run of the class-frequencies is usually sufficiently smooth to give a good
notion of the form of the ideal distribution.

Some common types of frequency-distribution

4.18 The forms presented by smoothly running sets of data are almost
endless in their variety, but among them we may notice a comparatively
small number of simple types. Such types also form a set into which
more complex distributions may often be analysed. For elementary

♦

Fig. 4.5. — ^An ideal symmetrical frequency-distrilnitioii

THEORY OF STATISTICS

purposes it is sufficient to consider four fundamental simple types, which
we shall call the symmetrical distribution, the moderately asymmetrical
or skew distribution,^ the extremely asymmetrical or J-shaped distribution
and the U-shaped distnbution. In the following sections we give some
examples of each of these types, together with a few more complex
distributions.

The symmetrical distribution

4,19 In this type the class-frequencies decrease to zero symmetrically
on either side of a central maximum. Fig. 4.5 illustrates the ideal form
of the distribution.

Being a special case of the more general type described under the
second heading, this form of distribution is comparatively rare. It

TABLE 4.7 — The frequency-distributions of statures for adult males born in England
Scotland, Wales and Ireland

As measurements are stated to have been taken to the nearest Jth of an inch, the
class-intervals are here presumably 561|-57^, 57-J|'-581|, and so on (cf. 4.9).
(See fig. 4.6.)

(Final Rei)ort of the Anthropometric Committee to the Bntish Association.) (Report, 1883, p. 256.)

Height without
shoes, inches

Number of men within said hmits of height
Place of birth —

England Scotland Wales Ireland

Total

57-

— —

58-

—

59-

—

60-

—

61-

62-

128

169

63-

320

394

64-

524

47 ~

669

65-

740

109

108

990

66- 1

881

139

145

1,223

67- 1

918

210

128

1,329

68-

886

210

1,230

69- 1

753

218

1,063

70-

473

115

646

71-

254

102

392

72- i

117

202

73-

74-

. —

75- i

—

76- 1

—

77- j

—

Total

6,194

1,304

741

346

8,585

^These two types, from their shape, are frequently referred to as “ humped,'
“ cocked hat,** " single peaked/* and so on.

FREQUENCY-DISTRIBUTIONS

occurs in the case of biometric, more especially anthropometric, measure-
ments, from which the following illustration is drawn, and is important
in much theoretical work. Table 4.7 shows the Irequency-dihtribation of
statures for adult males born in the British Isles, from data published by a
British Association Committee in 1883, the figures being given separately
for persons born in England, Scotland, Wales and Ireland, and totalled
in the last column. These frequency-distributions are approximately of
the symmetrical type. The frequency-polygon for the totals given b 3 ^
the last column of the table is shown in fig. 4 6. The student will notice
that an error of inch, scarcely appreciable in the diagram on its reduced
scale, IS neglected in the scale showm on the base-ime, the intervals being
treated as if they were 57-58, 58-59, etc. Diagrams should be drawn for
comparison showing, to a good open scale, the separate distributions for
England, Scotland, Wales and Ireland.

Fig. 4.6, — Frequency-distribution of stature for 8,585 adult males born in the Britlsb

Isles (Table 4.7)

The moderately asymmetrical (skew) distribution

4.20 In this case the class-frequencies decrease with markedly greater
rapidity on one side of the maximum than on the other, as in fig. 4.7 [a)
or (6). This is the most common of all smooth forms of frequency-
distribution, illustrations occurring in statistics from almost every source.
The distribution of birth-rates given in Table 4,1 is slightly asymmetrical.

THEORY OF STATISTICS

(a.)

Fig» 4.7. — Ideal distributions of the moderately asymmetrical form

The distribution of Australian marriages given in Table 4.8 (fig. 4.8)
is rather more asymmetrical and is of the type (a) of fig. 4.7. The
frequency attains its maximum for ages between 24 and 27 and then
tails off slowly. We have not drawn the tail of the curve, which is very
close to the ^-axis, for values of the variate above 58-5.

Table 4.9 and fig. 4.9 give a biological illustration, viz. the distribution
of fecundity (ratio of yearling foals produced to coverings) in mares.

TABLE 4.8. — Numbers of marriages contracted in Australia, 1907-14

Arranged according to the age of bridegroom in 3-year groups
(From S J. Pretoniis, “ Skew Bivanate Frequency Surfaces,” Bzomftrika, 1930, 22, 210) (See jfig. 4.8)

Age of bridegroom
(Ontral value of 3-year
range, in years)

Number of
mamages

Age of bridegroom
(Central value of 3-year
range, in years)

Number of
marriages

16-5

294

55-5

1,655

19-5'

10,995

58*5

1,100

22‘5

61,001

61*5 !

810

25*5

73,054

64-5

649

28-5

56,501

67*5

487

31-5

33,478

70-5

326

34-5

20,569

73 5

211

37*5

14,281

76*5

119

40*5

9,320

79-5

43-5

6,236

82*5

46-5

4,770

85-5

49-5

52*5

3,620

2,190

88*5

Total

301.785

FREQUENCY-DISTRIBUTIONS

- ms 2$-^ 2$5 315 M5 3T5 4(15 455 465 40’5 5Z5 55-5 56-5

Age of bi*iJegroom(gears)

Fig. 4.S.— Fie^iserk£3?«^st3rSbisti0ii of Aisstraliaa asarri&ges, classifM aceordlag to the bridegroom’s age (Table 4.S)

THEORY OF STATISTICS

The student should notice the difficulty of classihcation in this case :
the class-interval chosen throughout the middle of the range is 1 /15th,
blit the last interval is 29/30-1/' This is not a whole interval, but it
is more than a half, for all the cases of complete fecundity are reckoned
into the class. In the diagram (fig 4 9) it has been reckoned as a whole
class, and this gives a smooth distribution.

To take an illustration from meteorology, the distribution of barometer
heights at any one station over a period of time is, in general, asymmetrical,
the most frequent heights lying towaids the upper end of the range for
stations in England and Wales. Table 4.10 and fig. 4.10 show the dis-
tribution for daily observations at Greenwich during the years 1848-1926
inclusive.

The distributions of Tables 4.8-4.10 all follow more or less the t^vpe
of fig. 4.7 (a), the frequency taihng off, at the steeper end of the distribu-
tion, in such a way as to suggest that the ideal curve is tangential to the
base. Cases of greater asymmetry, suggesting an ideal curve that meets
the base (at one end) at a finite angle, even a right angle, as in fig. 4.7 (6),
are less frequent, but occur occasionally. The distribution of deaths
from scarlet fever, according to age, affords one such example of a more
asymmetrical kind. The actual figures for this case are given in Table
4.11 and illustrated by fig. 4.11 ; and it will be seen that the frequency
of deaths reaches a maximum for children aged 2 and under 3," the
number rising very rapidly to the maximum, and thence falling so slowdy

TABLE 4.9. — The frequency-distribution of fecundity, i.e. the ratio of the number of
yearling foals produced to the number of coverings, for brood-mares (racehorses)
covered eight times at least

(See fig. 4.9)

f Pearson, Lee and Moore, Phil. Trans , A, 1S99, 192, 303)

Fecundity

Number of
mares with
fecundity
between the
given limits

Fecundity

Number of
mares with
fecundity
between the
given limits

1 /30- 3 /30

17/30-19/30

315

3/30- 5/30

7*5

19 /30-2i /30

337

S/SO- 7/30

11*5

21/30-23/30

293*5

7/30- 9/30

21*5

23/30-25/30

204

9/30-11/30

25 /30~27 /30

127

11/30-13/30

104*5

27 /30-29/30

13/30-15/30

182

29 /30-1

15/30-17/30

271*5

Total

2000*0

FREQUENCY-DISTRIBUTIONS

Fig. 4.9. — Frequency-distribution of fecundity for brood-mares (Table 4.9)

that there is still an appreciable frequency for persons over 50 years of
age.

Asymmetrical curves are also said to be ''skew/' In Chapter 7 v/e
shall consider skewness at some length and discuss various ways of
measuring it. In particular we shall find that skewness has a sign, and
we may explain at this stage that the skewness is said to be positive if
the longer tail of the curve lies to the right, or negative if it lies to the
left ; e.g. the curve of fig. 4.8 has positive skewness, whilst those of figs. 4.9
and 4.10 have negative skewness.

The extremely asymmetrical, or J-shaped, distribution

4.21 In this type the class-frequencies run up to a maximum at one end
of the range, as in fig. 4.12.

This may be regarded as a limiting form of the previous distributiors,
and, in fact, the two cannot always be distinguished hy elementary methods
if the original data are not available. If, for instance, the frequencies oi
Table 4. 1 1 had been given by five-year intervals only, they would have run
322, 213, 70, 27, etc., thus suggesting that the maximuin number vl death«
occurred at the beginning of life, i.e. that the distribution was ! .'liapijd
It is only the analysis of deaths in the earlier years by one-\ iniervah
which shows that the frequencies reach a maximum in the liiird year and
that therefore the distribution is of the moderately asymmetrical type.
In practical cases no hard-and-fast rule can be drawn between the nioder-
ateiy and extremely asymmetrical types, any more than between the
asymmetrical and the symmetrical t 3 ^es.

THEORY OF STATISTICS

TABLE 4.10. — Barometric heights at Greenwich on alternate days from 1848 to 1926

(See fig. 4.10)

(Data from S. J. Pretonus, “ Skew Bxvanate Frequency Surfaces,” Btometnka^ 1930, 22, 154)

Barometric height

(Central value in

Number of days

(Central value m

Number of days

inches)

28 '35

29-65

3176

28 45

29-75

3700

28-55

29-85

3921

28-65

29-95

3749

28-75

30-05

2951

28-85

30-15

1951

28-95

189

30-25

1148

29-05

282

30-35

563

29-15

542

30*45

258

29-25

813

30-55 i

29-35

1233

30 65

29-45

29-55

1752

2333

30-75

Total

28,855

Fig. 4.10. — Barometric height at Greenwich on alternate days from i848"1926

{Table 4M}

FREQUENCY-DISTRIBUTIONS

TABLE 4.11. — ^The number of deaths brom scarlet fever at different ages in England

and Wales in 1933

{See fig 4 11)

(Data from Registrar-General's Statistical Review of England and Wales for 1933, Tables, Part I, Medical)

Age in years

Number of deaths

Number per year

213

42-6

10~

14-0

15-

5-4

20-

5-2

25-

3-4

30-

2 4

35-

2 2

40-

2-0

45-

1*2

50-

1-4

55-

1-0

60-

—

— .

65-

0-2

70-

0-2

75—

0*2

80-

—

Total

729

—

4.22 In economic statistics this form of distribution is particularly
characteristic of the distribution of wealth in the population at large, as
illustrated by income tax and house valuation returns, and the curve to
which it gives rise has been called the Pareto Hne/' after Vilfredo Pareto
who directed the attention of economists to it.

Such distributions may, of course, be a very extreme case of the last
type. It is difficult to say. But if the maximum is not absolutely at the
lower end of the range, it is very close thereto.

Official returns do not usually gi^e the necessary analysis of the
frequencies at the lower end of the range to enable the exact position of the
maximum to be determined ; and for this reason the data on which Table
4,12 is founded, though of course very unreliable, are of some interest. It
will be seen from the table and fig. 4.13 that with the given classification
the distribution appears clearly assignable to the present type, the number
of estates between zero and £100 in annual value being more than six times
as great as the number between £100 and £200 in annual value, and the
frequency continuously falling as the value increases. A close analysis of
the first class suggests, howevet, that the greatest frequency does not occur
actually at zero, but that there is a true maximum frequency for estates of
about £1 15 /- in annual value. The distribution might therefore be more

THEORY OF STATISTICS

correctly assigned to the second type, but the position of the greatest
frequency indicates a degree of skewness which is high even compared
with the skewness of fig. 4.11.

The type is more frequent in other classes of material than was at one
time thought. Distributions of deaths of centenarians afford an example,
and so, curiously enough, do deaths of infants unless the class-interval
is exceedingly fine — a matter of hours. The distribution may be obtained
by compiling the frequencies of the numbers of genera with 1, 2, 3, . . .
species in any biological group. Table 4.13 shows such a distribution for
the Chrysomelid beetles. Yule has also shown that it is characteristic
of the numbers of words used once, twice, thrice, etc., in a given work
and has used it in investigations into literary vocabularies.

The U-shaped distribution

4.23 This type exhibits a maximum frequency at the ends of the range
and a minimum towards the centre, as in fig. 4.14.

Age, in gears

Fig, 4,11.— Histogram of number of deaths from scarlet fever for various ai!es

(Table 4.11) ^

FREQUENCY-DISTRIBUTIONS 91 '

This IS a rare but interesting form of distribution, as it stands in some-
what marked contrast to the preceding forms. Table 4.14 and fig. 4.15
illustrate an example based on a considerable number of observations, viz.
the distribution of degrees of cloudiness, or estimated percentage of the sky
covered by cloud, at Greenwich in July.

For the purposes of the illustration we regard cloudiness as a variate
varying from complete overcastness to clear sky, the range being divided
into eleven equal parts.

It will be seen that a sky completely or almost completely overcast at
the time of observation is the most common, a practically clear sky comes
next, and the intermediates are more rare.

The remarks we made about the extreme end of the J-shaped dis-
tribution also apply to the U-shaped distribution. In particular cases it

Fig. 4.12. — An ideal distribution of the extremely asymmetrical form

may be that the grouping is too coarse to reveal the true character of the
frequency at the maxima, and if the data were more complete we might
discover that the two arms of the U in fact were bent over.

Truncated forms

4.24 The four types we have been considering sometimes occur in an
incomplete form. Certain limitations on the range of the variate may
result in a kind of truncation at one end or the other. Consider, for

^2 tHEOJtY OF STATISTICS

example, Table 4.15, p. 96. In obtaining these figures, twelve dice were
thrown and the occurrence of a 6 was called a success. At one throw there
could thus be any number of successes from 0 to 12. The dice were thrown
4096 times.

Fig, 4.13. — Frequency-distributloii of the annual values of certain estates in Englaiul
in 1715 ; 2,476 estates (Table 4.12)

Fig. 4,16 gives the frequency-polygon for this distribution. We can
picture it as a slightly skew distribution which has been cut off on the left
owing to the inadmissibility of negative values of the variate. Discon-
tinuous variates not infrequently give rise to this effect of truncation.

Complex distributioiis

4.25 Table 4.16 gives the number of male deaths within certain age-
limits for England and Wales in the years 1930-32.

Number of ohservatians per umt

THEORY OF STATISTICS

The histogram for these data is given in hg. 4.17. It will be seen that
the distribution has three maxima, one for each of the 0-5, the 20-25 and
the 70-75 age-groups.

Without looking too closely into this mortality curve we can see that
the high frequency at the beginning is undoubtedly due to the heavy
infantile death-rate. We can, if we choose, regard the distribution as

TABLE 4.12. — The numbers and annual values of the estates of those who had taken
part in the Jacobite rising of 1715

(See fig. 4.13)

(Compiled from Cosm’s “ Names of the Roman Cathohcs, Nonjurors, and others who Refused to take the Oaths to kts
laie Majesty King George, etc. ’ ; London, 1745. Figures of very doubtful absolute value See a note m Southey’s
'* Commonplace Book,” vol. 1, p, 573, quoted from the Memoirs of T Holhs)

Annual
value in
£100

Number of
estates

Annual
value in
£100

Number of
estates

0- 1

1726 5

17-18

1- 2

280

—

2- 3

140 5

20-21

3- 4

21-22

4- 5

46-5

22-23

5- 6

42*5

23-24

6- 7

29-5

—

7- 8

25-5

27-28

8- 9

18-5

—

9-10

31-32

10-11

U-5

—

11-12

9-5

39-40

12-13

—

13-14

3-5

45-46

14-15

—

15-16

3 1

48-49

16-17

Total

2,476

made up by the superposition of three others : a J-shaped distribution
for the lower years, a small one-humped distribution with its maximum
about the period 20-25 years, and a skew distribution for the higher
ages. This is an example of the fact we have already mentioned, that
a complex distribution can sometimes be analysed into simpler types.
In this particular case the analysis is likely to be of real service in actuarial
work and in investigations into the causes of death.

426 Finally, we give an example of a pseudo-frequency-distribution
of a type occasionally resorted to when the data can be classified according
to ’a characteristic which, though not strictly speaking measurable, can

FREQUENCY-BISTRIBUTIONS 95

nevertheless be graduated in an ordered sequence. Such a case anses
fairly often in psychological work.

A list of 100 words was read out to each of 11 subjects. Subsequently,
at 15-minute intervals, four fresh hsts were read out which contained 25
of the words in the original and 25 new words, the four taken together
accounting for the whole of the original 100. The subject had to say
whether these individual words were in the original list or not, and to
state whether he was certain, fairly sure, doubtful but inclined one way
or the other, or merely doubtful. The various phases of belief were
then allotted numbers, and ran from —3 (certainty that a word was not
in the original) through 0 (doubt, without inclination one way or the other)
to +3 (certainty that a word was in the original). The tabulation on p. 97
sets out the results for words in the original list (data reproduced by
permission from the records of the Department of Psychology, University
of St. Andrews).

TABLE 4.13. — Chrysomelidae (beetles). Numbers of genera with 1, 2, 3, . . . species

(Compiled by Dr. J, C. Willis, F.R.S. , ated from G. U. Yule, A Mathematical Theory of Evolution based
on the Conclusions of Dr. J. C. Willis,” Phil, Trans.^ B, 1924, 213, 85)

Species

8
9

20
21
22

(j^nera

215

14
5

15
8
9

6
8
6
6

5
4
2
3
1
3
3
3

Species

53
56

59
62
63

66
67
69

Genera

3
1
1
2
2
1

4
1
1
1
2
4
1
1
1
1
1
1
3
1
1
1
1
1
1
1

Species

77
79

84
87
89

93
110

114

115
128

132

133
146
163
196
217
227
264
327
399
417
681

Total

Genera

627

96 THEORY OF STATISTICS

TABLE 4.14. — The frequencies of estimated intensities of cloudiness at Greenwich
during the years 1890-1904 (excluding 1901) for the month of July

(See fig. 4.15)

(Data from Gertrude E. Pearse, Bwmdnka, 1928, 20A, 336)

Degrees of
cloudiness

Frequency

Degrees of
cloudiness

Frequency

676

148

129

320

Total

1,715

TABLE 4.15. — Twelve dice thrown 4,096 times, a throw of 6 points reckoned as a success

(See fig. 4.16)

(Weldon’s data, ated by F. Y. Edgeworth, Encyclopedia Britanmca^ 11th ed , 22, 39)

Number of successes .

7 and over

Total

Number of throws

447

1,145

1,181

796

380

115

4,096

Fig. 4.16. — ^Frequency polygon of successes with dice throwing (Table 4.15)

FREQUENCY-DISTRIBUTIONS

TABLE 4.16.— The number of male deaths in England and Wales for 1930-32

Classified by ages at death

(See fig 4.17)

(Data from Registrar-Generars Statistical Review of England and Wales, 1933, Text)

Age at death
(years)

Number of deaths

Age at death
(years)

Number of deaths

0- 5

97,290

55- 60

56,639

5-10

11,532

60- 65

68,103

10-15

7,305

65- 70

80,690

15-20

13,062

70- 75

84,041

20-25

16,741

75- 80

72,180

25-30

16,126

80- 85

45,094

30-35

15,673

85- 90

19,913

35-40

18,345

90- 95

5,145

40-45

23,778

95-100

767

45-50

33,158

100 and over

50-55

43,812

Total

i 729,442

Fig. 4.17. — Histogram of number of deaths at various ages (Table 4.16)

Words in the original list were classified as —

Possibly
either in
or out

Out

...... A.

Certain Fairly sure

Doubtful

Doubtful Fairly sure

Certain

+3 +2

—1 -2

-3

540 117

63 87

191

These results are very curious, and are borne out by other data of a
similar kind. In particular we see that there were more cases of certainty
about something which was not true than of doubt without incHnation.

THEORY OF STATISTICS

In this example we are clearly making some assumption in allotting
numbers to various degrees of belief ; but it would be impossible to
measure belief on a scale, and we have to do the best we can. The numbers
attached to the variate in such cases are not measures, but convenient
ordinals, like the numbers attached to kings of the same name. For
this reason a frequency diagram of such data can only give a very general
idea of their true nature.

SUMMARY

1. Data in which the individuals are specified by the numerical values
of a variable, or variate, may with convenience be arranged in a table
which gives the frequency lying within successive, preferably equal,
ranges of the variable. Such an arrangement is called a frequency-
distribution.

2. The frequency-distribution can be represented diagrammatically by
means of a frequency-polygon or a histogram.

3. The histogram is particularly appropriate to cases in which the
frequency changes rapidly or the class-intervals are not all of the same
width.

4. As the width of the class-intervals becomes smaller, the frequency-
polygon or the histogram may be imagined to approach a smooth curve,
which is called the frequency-curve.

5. A large number of frequency distributions occurring in practice
fall into four t 3 q>es : the symmetrical, the moderately asymmetrical or
skew, the extremely asymmetrical or J-shaped and the U-shaped types.
Certain other distributions can be analysed into constituents each of
which belongs to one of these types.

EXERCISES

4.1 If the diagram fig. 4.6 is redrawn to scales of 300 observations per
interval to the inch and 4 inches of stature to the inch, what is the scale
of observations to the square inch ?

If the scales are 100 observations per interval to the centimetre and 2
inches of stature to the centimetre, what is the scale of observations to the
square centimetre ?

4.2 If fig. 4.10 is redrawn to scales of 900 days to the inch and 0 • 3 inch of
barometric height to the inch, what is the scale of observations to the
square inch ?

FREQUENCY-DISTRIBUTIONS

If the scales are 400 days to the centimetre and 0*1 inch of barometric
height to the centimetre, what is the scale of observations to the square
centimetre ^

4.3 If a frequency-polygon be drawn to represent the data of Table
4.1, what number of observations will the polygon show between birth-
rates of 16-5 and 17*5 per thousand, instead of the true number 89 ?

4.4 If a frequency-polygon be drawn to represent the data of Table 4.6,
what number of observations will the polygon show between head-breadths
5*95 and 6*05, instead of the true number 236 ?

4.5 Draw frequency^-polygons or histograms, as the case seems to require,
for the following distributions, and assign them to the four types w^e have
enumerated in 4.18 —

(a) Size of firms in the food, drink and tobacco trades of Great Britain

The table shows the number of firms employing on an average certain numbers

of persons —

(Final Report of the Fourth Census of Production, 1930, Part III)

Size of firm (av-
erage numbers 11-24 25-49 50- 100- 200- 300- 400- 500- 750- lOOO- 1,500 ^ ,

employed) 99 199 299 399 499 74 9 999 1,499 and over

Numberof firms 2,245 1,449 771 439 164 75 36 54 31 23 29 5,316

(b) The percentages of deaf-mutes among children of parents one of whom at least was a
deaf-mute, for marriages producing five children or more

(Compiled from matenal m " Marnagis of the Deaf in Americaf* ed E Fay, Volta Bureau, Washington, 1898)

Percentage

deaf-mutes

Number of
famihes

Percentage

deaf-mutes

Number of
families

0-20

220

60- 80

5*5

20-40

20*5

80-100

40-60

Total

273

(Mercer and Hall, “ The Expenmental Error of Field TiiaE,’* Journ. Agr. Science, 4, 1911, 107)

Yield of gram m

pounds per gJath ^.s 3.0 3-2 3-4 3-6 3-S 4-0 4-2 4-4 4-6 4 8 5-0 5-2 Total
acre (uentrai

value of range)

Number of plots . 4 15 20 47 63 78 88 69 59 35 10 8 4 500

100

THEORY OF STATISTICS

(d) The frequencies of different numbers of petals for three series of ranunculus bulbosus

(H de Vnes, Ber deuisch bot Ges , Bd. 12, 1894, qv for details)

Number

Frequency

of petals

Senes A

Senes B

Series C

312

345

133

—

Total

337

380

222

4.6 A number of perfectly spherical balls, all of the same material, give a
symmetrical distribution when classified according to their diameters.
Show that, if they are classified according to their weights, their frequency-
distribution will be positively skew towards the higher weights.

Table to Exercise 4.6

The frequency-distribution of weights for adult males born in England, Scotland, Wales
and Ireland (loc. cit, Table 4.7)

Weights were taken to the nearesf* pound, consequently the true class-intervals are

89 -5-99 -5, 99 -5-109 -5, etc.

Weight
in lb

Number

England

of men within given limits of
weight. Place of birth —

Scotland Wales Ireland

Total

90-

100-

no-

133

152

120-

338

390

130-

694

867

140-

1,240

173

153

1,623

150-

1,075

255

178

1,559

160-

881

275

134

1,326

170-

492

168

102

787

180-

304

125

476

190-

174

263

200-

107

210-

220-

—

230-

—

240-

—

250-

—

260-

—

270-

—

280-

—

Total

5,552

1,212

738

247

7,749

FREQUENCY-DISTRIBUTIONS

lOI

In the light of this result compare the distributions of Table 4.7 with the
distributions of the table on the previous page.

4.7 Toss a coin six times and note the number of heads. Repeat the
experiment 100 times or more, and draw a frequency-polygon of your
results classified according to the number of heads at each throw.

4.8 Find the frequency-distribution of 200 bars of a waltz by Strauss
classified according to the number of notes in the principal melody in the
treble clef of each bar, and compare it with a similar distribution from
modem waltzes.

4.9 Examine qualitatively the effect on the distribution of Table 4.8
of an allowance for the fact that minors tend to overstate their age when
marr5dng.

4.10 The distribution of a herd of cows classified according to the quantity
of milk produced by each cow per week is symmetrical. The distribution
of the same herd classified according to the amount of butter-fat produced
by each cow per week is negatively skew towards the lower quantities.
Suggest a possible explanation for this fact.

CHAPTER FIVE

AVERAGES AND OTHER MEASURES OF

LOCATION

The principal characteristics of frequency-distributions

5.1 The condensation of data into a frequency-distribution is a first
and necessary step in rendering a long series of observations compre-
hensible. But for practical purposes it is not enough, particularly w^hen
we want to compare two or more different series. As a next step we wish
to be able to define quantitatively the characteristics of a frequency-
distribution in as few numbers as possible.

5.2 It might seem at first sight that very difficult cases of comparison
of two distnbutions could arise in which, for example, we had to contrast
a symmetrical distribution with a J-shaped distribution. In practice,
however, we rarely have to deal with such a case. Distributions drawn
from similar material are usually of similar form — as, for instance, when
we wish to compare the distnbutions of stature in two races of man, or
the birth-rates in English registration districts in two successive decades,
or the numbers of wealthy people in two different countries. The practical
use of the various statistical quantities which we shah discuss in this
and the next two chapters is based on this fact.

5.3 There are two fundamental characteristics in which similar frequency-
distributions may difer —

(1) They may differ markedly in position, i.e. in the value of the variate
round which they centre, as in fig. 5.1, A.

(2) They may differ in the extent to which the observations are dis-
persed about the central value. Figs. 5.1, B and C, show cases in which
distributions differ in dispersion only, and in both dispersion and position,
respectively.

To these two characteristics we may add a third group of less import-
ance, comprising differences in skewness, peakedness, and so on.

Measures of the first character, i.e. position or location, are generally
known as averages. Measures of the second are termed measures of
dispersion. Measures of the properties in the third group have each
their appropriate name, which we shall give when we come to consider
them in detail.

102

AVERAGES

103

The present chapter deals only with averages. Chapter 6 deals with
measures of dispersion, whilst Chapter 7 deals with the remaining
quantities.

Dimensions of an average

5.4 In whatever way an average is defined, it may be as well to note
it is merely a certain value of the variable, and is therefore necessarily
of the same d%menstons as the variable : i.e. if the variable be a length,
its average is a length ; if the variable be a percentage, its average is a
percentage ; and so on. But there are several different ways of approxi-
mately defining the position of a frequency-distribution — that is, there

(t) (2)

(I)

are several different forms of average, and the question therefore arises.
By what criteria are we to judge the relative merits of different forms ?
What are, in fact, the desirable 'properties for an average to possess ?

Desiderata for a satisfactory average

5.5 (a) In the first place, it almost goes without saying that an average
should be rigidly defined, and not left to the mere estimation of the
observer. An average that was merely estimated would depend too
largely on the observer as well as the data.

(b) An average should be based on aU the observations made. If not,
it is not reaUy a characteristic of the whole distribution.

(c) It is desirable that the average should possess some simple and
obvious properties to render its general nature readily comprehensible :
an average should not be of too abstract a mathematical character.

104

THEORY OF STATISTICS

{d) It is, of course, desirable that an average should be calculated with
reasonable ease and rapidity. Other things being equal, the easier
calculated is the better of two forms of average. At the same time
great weight must not be attached to mere ease of calculation, to the
neglect of other factors.

(e) It is desirable that the average should be as little affected as may
be possible by what we have termed fluctuations of sampling. If different
samples be drawn from the same material, however carefully they may
be taken, the averages of the different samples will rarely be quite the
same, but one form of average may show much greater differences than
another. Of the two forms, the more stable is the better. The full
discussion of this condition must, however, be postponed to a later section
of this work (Chap. 18).

(/) Finally, by far the most important desideratum is this, that the
measure chosen shall lend itself readily to algebraical treatment. If,
e.g., two or more series of observations on similar material are given,
the average of the combined series should be readily expressed in terms
of the averages of the component series ; if a variable may be expressed
as the sum of two or more others the average of the whole should be
readily expressed in terms of the averages of its parts. A measure for
which simple relations of this kind cannot be readily determined is likely
to prove of somewhat limited application.

5.6 There are three forms of average in common use, the arithmetic
mean, the median and the mode, the first named being by far the most
widely used in general statistical work. To these may be added the
geometric mean and the harmonic mean, more rarely used, but of service
in special cases. We will consider these in the order named.

The arithmetic mean

5.7 The arithmetic mean of a series of values of a variable X^, X^,
Xa, . . . Xn, N in number, is the quotient of the sum of the values by
their number. That is to say, if M be the arithmetic mean,

^=^(^l+^*+^3+ • • • +Xn)

The arithmetic mean is also denoted by placing a bar over the variate
symbol, so that we may also write—

AVERAGES

105

To express these formulae more briefly by the use of the summation
symbol S,

= (5.1)

The word mean or average alone, without qualification, is very generally
used to denote this particular form of avc ge ; that is to say, when anyone
speaks of ‘‘ the mean or the average of a series of observations, it
may, as a rule, be assumed that the arithmetic mean is meant.

5.8 It is evident that the arithmetic mean fulfils the conditions laid
dowm in [a) and [h) of 5.5, for it is rigidly defined and based on all the
observations made. Further, it fulfils condition (c), for its general nature
is readily comprehensible. If the wages-bill for N w^orkmen is £-P, the
arithmetic mean wage, P jN pounds, is the amount that each would
receive if the whole sum available were divided equally between them :
conversely, if we are told that the mean wage is £M, we know this means
that the wages-bill is NM pounds. Similarly, if iV families possess a total
of C children, the mean number of children per family is C fN — the number
that each family would possess if the children were shared unifomaly.
Conversely, if the mean number of children per family is M, the total
number of children in N families is NM. The arithmetic mean expresses,
m fact, a simple relation between the whole and its parts.

The mean is also satisfactory as regards conditions {e) and {/), but we
shall have to defer proof of this statement for the present.

. Calculation of the. arithmetic mean

5.9 As regards condition {d), simplicity of calculation, the mean takes
a high place. In the cases just cited, it will be noted that the mean is
actually determined without even the necessity of determining or noting
all the individual values of the variable : to get the mean wage we need not
know the wages of every hand, but only the wages*bill ; to get the mean
number of children per family we need not know the number in each
family, but only the total. If this total is not given, but we have to deal
with a moderate number of observations — so few (say 30 or 40) that it is
hardly worth while compiling the frequency-distribution — the arithmetic
mean is calculated directly as suggested by the definition, i.e. all the values
observed are added together and the total divided by the number of
observations.

5.10 But if the number of observations be large, the process of adding
together all the values of the variate may be prohibitively lengthy. It
may be shortened considerably by forming the frequency-table and treat-
ing all the values in each class as if they were identical with the mid-value
of the class-interval, a process which in general gives an approximation
that is quite sufldciently exact for practical purposes if the class-interval

io6

THEORY OF STATISTICS

has been taken moderately smail. In this process each class-frequency
is multiplied by the mid-value of the interval, the products added together,
and the total divided by the number of observations. Tf / denote the
frequency of any class, X the mid- value of the corresponding class-interval,
the value of the mean so obtained may be written —

. . . (5.2)

5.11 But this procedure is still further abbreviated in practice by the
following artifices : (1) The class-interval is treated as the unit of measure-
ment throughout the arithmetic , (2) the difference between the mean
and the mid-value of some arbitrarily chosen class-interval is computed
instead of the absolute value of the mean
If /I be the arbitrarily chosen value and

(5,3)

then

S(/Z)=2(/^)+S(/C)

or, since ^ is a constant,

.... (5.4)

The calculation of S(/Z) is therefore replaced by the calculation of
S(/g). The advantage of this is that the class-frequencies need only be
multiplied by small integral numbers ; for A being the mid-value of a
class-interval, and X the mid-value of another, and the class-interval being
treated as a unit, the g's must be a series of integers proceeding from zero
at the arbitrary origin A. To keep the values of ^ aa small as possible, A
should be chosen near the middle of the range.

It mBY be mentioned here that or for the grouped dis-

tribution, is sometimes termed the first moment of the distribution about
the arbitrary origin A,

Example 5,1. — As an example, let us find the arithmetic mean of the
heights in the ''total” column of Table 4.7. In this case the class-interval
is a unit (1 inch), so the value of ilf—A is given directly by dividing S(/£)
by N. The student must notice that, measures having been -made to the
nearest eighth of an inch, the mid-values of the intervals are 57^, 58*]^,
etc., and not 57*5, 58*5, etc.

AVERAGES

107

Calculation of the arithmetic mean stature of male adults in the British Isles from the

figures of Table 4.7, p. 82

(1)

Height,

inches

(2)

Frequency

(3)

Deviation
from arbitrary
\ alue A

(4)

Product

/£

57-

-10

- 20

58-

- 9

- 36

59-

- 8

- 112

60-

- 7

~ 287

61-

- 6

- 498

62-

169

- 5

- 845

63-

394

- 4

-1576

64-

669

- 3

-2007

65-

990

- 2

-1980

66-

1223

- 1

-1223

67-

1329

-8584

68-

1230

“f 1 j

1230

69-

1063 1

-f 2

2126

70-

646

-f 3

1938

71-

392

+ 4

1568

72-

202

+ 5

1010

73-

-f 6

474

74-

-f 7

224

75-

-f 8

128

76-

-h 9

77-

-fio

Total

8585

—

+8763

5:(/g) = 48,763-8,584 = 4179
179

M—A = H 40*02 class-intervals or inches.

8,585

M = 67 •^40*02 = 67*46 inches.

3,12 As calculations of the mean constantly have to be made, the
student should familiarise himself with the process we have just illustrated,
and note that a check can always be effected on the arithmetic in the
following way —

Since /(g+1) =/g4/

2;/(g41);=2(/g)4Si/)

Sl/(£41):-Wr)=S(/^

= Total frequency

Hence, if we tabulate the values of /(^41) well as those of and find
their totals, the difference must, if the arithmetic is correct, be equal to
the total frequency.

io8

THEORY OF STATISTICS

5.13 It will be evident that a classification by unequal intervals is,
at best, a hindrance in the calculation of the mean, and the use of an
indefinite interval at the end of the distribution renders exact calculation
impossible. The following example illustrates the calculation for unequal
class-intervals and the arithmetical check to which we have just referred.

Example 5.2. — Data from Table 4.11, page 89. What is the average
age at death from scarlet fever ?

Here there is a change of the class-interval at the five-year point. We
take a year to be the unit, and the centre of the interval 5-10 years as an
arbitrary origin, which means that A =7-5 years.

Calculation of the arithmetic mean age of persons dying from scarlet fever in the
United Kingdom in 1933 (Table 4.11, p. 89)

Age

Years

Frequency

Deviation from A

/£

/(g+i)

—7

- 112

- 96

-6

- 414

- 345

-5

~ 445

- 356

-4

- 296

- 222

-3

- 222

- 148

213

-1489

-1167

213

10-

350

420

15-

270

297

20-

390

416

25-

340

357

30-

300

312

35-

330

341

40-

350

t 360

45-

40 ;

240

246

50-

315

322

55-

250

255

60-

—

65-

70-

75-

1 71

Total

729

—

+ 3330

+ 3737

Hence,

and

S(/g) =3330-1489=1841

=.3737-1167=2570

and the difference 2570—1841=729, as it should.

Hence,

M -^=—=2-525 years
/

M=7-5 +2-525==10-025 years

and

AVERAGES

109

5.14 We return again below, m 5.16 (c), to the question of the errors
caused by the assumption that all values within the same interval may be
treated as approximately the mid- value of the interval. It is sufficient to
say here that the error is in general very small and of uncertain sign for a
distribution of the symmetrical or only moderately asymmetrical type,
provided of course, the class-interval is not large. In the case of the

J-shaped or extremely asymmetrical distribution however, the error is
evidently of definite sign, for in aU the intervals the frequency is piled up
at the limit lying towards the greatest frequency, i.e. the lower end of the
range in the case of the illustrations given in Chapter 4, and is not evenly
distributed over the interval. In distnbutions of such a type the intervals
must be made very small indeed to secure an approximately accurate value
for the mean. The student should test for himself the effect of different
groupings in two or three different cases, so as to get some idea of the degree
of inaccuracy to be expected.

5.15 If a diagram has been drawn representing the frequency-distribution,
the position of the mean may conveniently be indicated by a vertical
through the corresponding point on the base. In a moderately asym-
metrical distribution the mean lies on the side of the greatest frequency
toward^ the longer tail ” of the distribution : M in fig. 5.2 shows the

Fig. 5.2. — Mean M, median Mi and mode Mo of the ideal moderately asymmetrical

distribution

position of the mean in an ideal distribution. In a symmetrical distribu-
tion the mean coincides with the centre of symmetry. The student should
mark the position of the mean in the diagram of every frequency-dis-
tribution that he draws, and so accustom himself to thinking of the mean
not as an abstraction, but always in relation to the frequency-distribution
of the variable concerned.

THEORY OF STATISTICS

Properties of the arithmetic mean

5.16 The following are important properties of the arithmetic mean,
and the examples illustrate the facihty of its algebraic treatment —

(a) The sum of the deviations from the mean, taken with their proper
signs, IS zero.

This follows at once from equation (5 4) * for if M and A are identical,
evidently S(/^) must be zero.

(b) If a series of N observations of a variable X consist of, say, two
component series, the mean of the whole senes can be readily expressed
in terms of the means of the two components. For if we denote the values
in the first series by X-^ and m the second series by Xg,

S(Z)=2(A\)+S(Z2)

that is, if there be observations in the first senes and in the second,
and the means of the two series be Mo, respectively,

.... (5.5)

For example, we find from the data of Table 4.7,

Mean stature of the 346 men born in Ireland =67 *78 inches
„ „ „ 741 „ „ Wales =66*62 „

Hence the mean stature of the 1087 men born in the two countries is given
by the equation

1087M=(346x67*78)+(741 y 66*62)
that is, M =66 • 99 inches.

It is evident that the form of the relation (5.5) is quite general : if
there are r series of observations Xj, X^, • . . X^, the mean M of the
whole series is related to the means My, ... of the component
series by the equation

NM . . . +NrMr . . (5.6)

For the convenient checking of arithmetic, it is useful to note that, if the
same arbitrary origin A for the deviations g be taken in each case, we must
have, denoting the component series by the subscripts 1 , 2, ... r as before,

Zifi) • • ■ +S(/rQ . • ■ (5.7)

The agreement of these totals accordingly checks the work.

As an important corollary to the general relation (5.6), it may be noted

AVERAGES

III

that the approximate value for the mean obtained from any frequency-
distribution is the same whether we assume (1) that all the values in any
class are identical with the mid-value of the class-interval, or (2) that the
mean of the values in the class is identical with the mid-value of the class-
interval.

(c) The mean of all the sums or differences of corresponding observa-
tions in two series (of equal numbers of observations) is equal to the sum
or difference of the means of the two series.

This follows almost at once For if

That is, if M, Mg be the respective means,

(5.8)

Evidently the form of this result is again quite general, so that if

. . . ±x,

, ±Mr . . . (5.9)

As a useful illustration of equation (5.8), consider the case of measurements
of any kind that are subject (as indeed all measures must be) to greater or
less errors. The actual measurement X in any such case is the algebraic
sum of the true measurement and an error X.^. The mean of the actual
measurements M is therefore the sum of the true mean and the
^thmetic mean of the errors Mg. If, and only if, the latter be zero, will
the observed mean be identical with the true mean. Errors of grouping
(5.14) are a case in point.

Tl^ Median

5.17 The median may be defined as the middlemost or central value
of the variable when the values are ranged in order of magnitude, or as the
value such that greater and smaller values occur with equal frequency. In
the case of a frequency-curve, the median may be defined as that value of
the variable the vertical through which divides the area of the curve into
two equal parts, as the vertical through Mi in fig. 5.2.

The median, like the mean, fulfils the conditions (6) and (c) of 5.5, seeing
that it is based on all the observations made, and that it possesses the
simple property of being the central or middlemost value, so that its
nature is obvious.

5.18 But the definition does not necessarily lead in all cases to a deter-
minate value. If there be an odd number of different values of X observed,
say 2n + l, the l)th in order of magnitude is the only value fulfilling

II2

THEORY OF STATISTICS

the definition. But if there be an even number, say 2n different values,
any value between the nth. and (w + l)th fulfils the conditions. In such
a case it appears to be usual to take the mean of the nth and (r^+l)th
values as the median, but this is a convention supplementary to the
definition.

5.19 It should also be noted that in the case of a discontinuous variable
the second form of the definition in general breaks down : if we range
the values in order there is always a middlemost value (provided the
number of observations be odd), but there is not, as a rule, any value such
that greater and less values occur with equal frequency. Thus, in Table
4.2 we see that 45 per cent of the poppy capsules had 12 or fewer stigmatic
rays, 55 per cent had 13 or more ; similarly, 61 per cent had 13 or fewer
rays, 39 per cent had 14 or more. There is no number of rays such that
the frequencies in excess and defect are equal. In the case of the butter-
cups of Exercise 4.5 [d), page 100, there is no number of petals that even
remotely fulfils the required condition. An analogous difficulty may arise,
it may be remarked, even in the case of an odd number of observations of a
continuous variable if the number of observations be small and several of
the observed values identical.

The median is therefore a form of average of most uncertain meaning in
cases of strictly discontinuous variation, for it may be exceeded by 5, 10,
15 or 20 per cent only of the observed values, instead of by 50 per cent :
its use in such cases is to be deprecated, and is perhaps best avoided in any
case, whether the variation be continuous or discontinuous, in which small
series of observations have to be dealt with.

Determination of the median

5.20 When aU the values of the variate are given and the total frequency
is small, the median can be determined by inspection as the middlemost
value or, if there is no such value, as the mean of the two middlemost
values. When the distribution is given as a frequency-distribution,
however, a -certain amount of approximation is necessary, as in the case
of the calculation of the mean.

For the frequency-distribution of a continuous variable a sufficiently
approximate value of the median can be obtained by interpolation. If
the total frequency is large it is sufficient to assume that the values in each
class are uniformly distributed throughout the interval.

Example 5.3. — Let us determine the median of the distribution whose
mean we found in Example 5.1. The work may be indicated thus —

Half the total number of observations (8585) . = 4292-5

Total frequency under 66 -fl inches . . = 3589

Difference = 703-5

Frequency in next interval , . . . = 1329

AVERAGES

Hence we take the median to be —

703*5

1329

= 67*47 inches

The difference between the median and mean in this case is therefore
only about one-hundredth of an inch.

Example 5.4. — To find the median of the distribution of Example 5.2.

Half the total number of observations . . =364*5

Total frequency under 5 years . . = 322

Difference . = 42*5

Frequency in next interval , . . . = 213

Hence we take the median to be —

42*5

213

= 6 years

Here the median is very far from coinciding with the mean.

Graphical determination of the median

5.21 Graphical interpolation may, if desired, be substituted for arith-
metical interpolation. Taking the figures of Example 5.1, we see that
the number of men with height less than 65 H is 2366, less than 66
is 3589, less than 67|| is 4918, and less than 68-J| is 6148.

Plot the numbers of men with height not exceeding each value of X
to the corresponding value of X on squared paper, to a good large scale,
as in fig. 5.3, and draw a smooth curve through the points thus obtained,
preferably with the aid of one of the curves,"' splines or flexible curves
sold by instrument-makers for the purpose. The point at which the
smooth curve so obtained cuts the honzontal line corresponding to a
total frequency N /2 =4292 *5 gives the median. In general the curve is
so flat that the value obtained by this graphical method does not differ
appreciably from that calculated arithmetically (the arithmetical process
assuming that the curve is a straight line between the points on either
side of the median) ; if the curvature is considerable, the graphical value
— assuming, of course, careful and accurate draughtsmanship — ^is to be
preferred to the arithmetical value, as it does not involve the crude
assumption that the frequency is uniformly distributed over the interval
in which the median lies.

THEORY OF STATISTICS

II4

Fig. 5.3. — Determination of the median by graphical interpolation
Comparison of the mean and the median

5.22 If we adopt the convention that the median of an even number
of observations is midway between the two central values, both the
mean and the median satisfy the first three of the desiderata we enumerated
in 5.5 , that is to say, they are rigidly defined, based on all the observa-
tions, and are readily comprehensible. In the remaining three, however,
they differ considerably.

5.23 As regards ease of calculation, the median has distinct advan-
tages over the mean. i

Whether the stability of the median under fluctuations of sampling
is greater than that of tljie mean depends to some extent on the form
of the distribution which is being sampled. In general, the mean is
the more stable, but case's occur in which the median is preferable (cf.

5.24 {d) below, and Chap. 18).

When, however, the ease of algebraical treatment of the two forms
of average is compared, the superiority hes wholly on the side of the mean.

AVERAGES

II5

As was shown in 5.16, when several series of observations are combined
into a single series, the mean of the resultant distribution can be simply
expressed in terms of the means of the components. Expression of
the median of the resultant distribution in terms of the medians of the
components is, however, not merely complex and difficult, but usually
impossible : the value of the resultant median depends on the forms of the
component distributions, and not on their medians alone. If two sym-
metrical distributions of the same form and with the same numbers of
observations, but with different medians, be combined, the resultant median
must evidently (from symmetry) coincide with the resultant mean, i.e. lie
half-way between the means of the components. But if the two com-
ponents be asymmetrical, or (whatever their form) if the degrees of
dispersion or numbers of observations in the two series be diferent, the
resultant median will not coincide with the resultant mean, nor with
any other simply assignable value. It is impossible, therefore, to give
any theorem for medians analogous to equations (5.5) and (5.6) for
means. It is equally impossible to give any theorem analogous to
equations (5.8) and (5.9) of 5.16. The median of the sum or difference
of pairs of corresponding observations in two series is not, in general,
equal to the sum or difference of the medians of the two series ; the
median value of a measurement subject to error is not necessarily identical
with the true median, even if the median error be zero, i.e. if positive
and negative errors be equally frequent.

5.24 These limitations render the applications of the median in any
work in which theoretical considerations are necessary comparatively
circumscribed. On the other hand, the median may have an advantage
over the mean for special reasons.

(a) It is very readily calculated ; a faetdrto which, however, as already
stated, too much weight ought not to be attached.

(b) It is readily obtained, without the necessity of measuring all the
objects to be observed, in any case in which the objects can be arranged
in order of magnitude. If, for instance, a number of men be ranked in
order of stature, the stature of the middlemost is the median, and he
alone need be measured. (On the other hand, it is useless in the cases
cited at the end of 5.8 ; the median wage cannot be found from the
total of the wages-bill, and the total of the wages-bill is not known when
the median is given.)

(c) It is sometimes useful as a makeshift, when the observations are
so given that the calculation of the mean is impossible, owing, e.g., to a
final indefinite class.

(d) The median may sometimes be preferable to the mean, owing to
its being less affected by abnormally large or small values of the variable.
The stature of a giant would have no more influence on the median
stature of a number of men than the istature-of any other man whose

ii6

THEORY OF STATISTICS

height is only just greater than the median. If a number of men enjoy
incomes closely clustering round a median of £500 a year, the median
will be no more affected by the addition to the group of a man with an
income of £50,000 than by the addition of a man with an income of £5,000,
or even £600. If observations of any kind are liable to present occasional
greatly outlying values of this sort (whether real, or due to errors or
blunders), the median will be more stable and less affected by fluctuations
of sampling than the arithmetic mean (cf. Chap. 18).

{e) It may be added that the median is, in a certain sense, a particu-
larly real and natural form of average, for the object or individual that
is the median object or individual on any one system of measuring the
character with which we are concerned will remain the median on any
other method of measurement which leaves the objects in the same relative
order. Thus a batch of eggs representing eggs of the median price,
when paces are reckoned at so much per dozen, will remain a batch
representing the median price when prices are reckoned at so many eggs
to the shilling.

The mode

5.25 The mode is the value of the variable corresponding to the maximum
of the ideal curve which gives the closest possible fit to the actual dis-
tribution. It represents the value which is most frequent or typical,
the value which is, in fact, the fashion {la mode)}- The mode is sometimes
denoted by writing the sign over the variate symbol, e.g. X means
the mode of the values Zg, . . . Z^.

There is evidently something anticipatory about this definition, for
we have not yet defined what we mean by closest possible fit.” For
the present the student must content himself with intuitive ideas on this
head. Nor have we given a method of finding the curve of closest fit,
which wwld be a necessary preUminary to ascertaining the mode.

5.26 It is, in fact, difficult to determine the mode for such distributions
as arise in practice, particularly by elementary methods. It is no use
giving merely the mid- value of the class-interval into which the greatest
frequency falls, for this is entirely dependent on the choice of the scale
of class-intervals. It is no use making the class-intervals very small
to avoid error on that account, for the class-frequencies will then become
small and the distribution irregular. What we want to arrive at is the
mid- value of the interval for which the frequency would be a maximum,
if the intervals could be made indefinitely small, and at the same time
the number of observations be so increased that the class-frequencies

^ Unless we state expressly to the contrary, we shall be thinking of single-humped
distributions in talking of “ the " mode. When the distnbution is of the complicated
form of fig. 4,17 there may be more than one mode Such distributions are therefore
sometimes called multimodal. The mean and the median are still unique for such
distributions.

AVERAGES

II7

should run smoothly. As the observations cannot, in a practical case,
be indefinitely increased, it is evident that some process of smoothing
out the irregularities that occur in the actual distribution must be adopted,
in order to ascertain the approximate value of the mode. But there is
only one smoothing process that is really satisfactory, in so far as every
observation can be taken into account in the determination, and that
is the method of fitting an ideal frequency-curve of given equation to
the actual figures. The value of the variable corresponding to the
maximum of the fitted curve is then taken as the mode, in accordance
with our definition. The determination of the mode by this — the only
strictly satisfactory — method must, however, be left to the more advanced
student. The methods of curve-fitting which we shall discuss in Chapter 15
are not appropriate to the fitting of frequency-curves, but we give an
approximate method which is of use in certain cases in 25.21.

Empirical relation between mean, median and mode
5.27 For a symmetrical distribution, mean, median and mode coincide,
as will be evident on a little consideration. For other distributions as
a rule, they do not. Fig. 5.2 shows the position of the three in a
moderately skew distribution.

There is an approximate relation between mean, median and mode
which appears to hold good with surprising closeness for moderately
asymmetrical distributions, approaching the ideal type of hg. 4.7, and it
is one that should be borne in mind as giving — ^roughly, at all events —
the relative values of these three averages for a great many cases with
which the student will have to deal. It is expressed by the equation

Mode = Mean— 3(Mean— Median)

That is to say, the median lies one-third of the distance mean to mode
from the mean towards the mode. The student will find it easy to
remember this relation if he notes that mean, median and mode occur
in the same order (or the reverse order) as in the dictionary, and that the
median is nearer to the mean, also as in the dictionary.

The following table gives the true mode and the mode calculated in
accordance with the above formula for certain skew distributions of the
type of fig. 4.10 —

Comparison of the approximate and true modes in the case of five distributions of the
height of the barometer for daily observations at the stations named

(Distnbutions given by Karl Pearson and Alice Lee, Pktl. Trans , A. 1897, 190 , 423)

Station

Mean

Median

Approximate

Mode

True Mode

Southampton .

29-981

30-000

30-038

30-039

Xxindonderry .

29-891

29-915

29-963

29*960

Carmarthen

29-952

29-974

30-018

30-013

Glasgow .

29-886

29-906

29-946

29-967

Dundee .

29*870

29-890

29-930

29-951

Il8 THEORY OF STATISTICS

It will be seen that the true and approximate values are extremely
close, except in the case of Dundee and Glasgow, where the divergence
reaches two-hundredths of an inch.

5.28 Summing up the preceding paragraphs, we may say that the mean
IS the form of average to use for all general purposes ; it is simply cal-
culated, its value is nearly always determinate, its algebraic treatment is
particularly easy, and in most cases it is rather less affected than the
median by errors of sampling. The median is, it is true, somewhat more
easily calculated from a given frequency-distribution than is the mean ;
it is sometimes a useful makeshift, and in a certain class of cases it is
more and not less stable than the mean ; but its use is undesirable in
cases of discontinuous variation, its value may be indeterminate, and its
algebraic treatment is difficult and often impossible. The mode, finally,
is a form of average hardly suitable for elementary use, owing to the
difficulty of its determination, but at the same time it represents an
important value of the variable. The arithmetic mean should invariably
be employed unless there is some very definite reason for the choice of
another form of average, and the elementary student will do very well
if he limits himself to its use. Objection is sometimes taken to the use
of the mean in the case of asymmetrical frequency-distributions, on the
ground that the mean is not the mode, and that its value is consequently
misleading. But no one in the least degree familiar with the manifold
forms taken by frequency-distributions would regard the two as in general
identical ; and while the importance of the mode is a good reason for
stating its value in addition to that of the mean, it cannot replace the
latter. The objection, it may be noted, would apply with almost equal
force to the median, for, as we have seen (5.27), the difference between
mode and median is usually about two-thirds of the difference between
mode and mean.

The geometric mean

5.29 The geometric mean G of a series of values X^, X3. . . . X^

is defined by the relation

. , . Z^)Viv . . . (5.10)

The definition may also be expressed in terms of logarithms —

logG=is(log Z) .... (5.11)

that is to say, the logarithm of the geometric mean of a series of values
is the arithmetic mean of their logarithms.

The geometric mean of a given series of quantities is always less than
their arithmetic mean ; the student will find a proof in most textbooks

of algebra. The magnitude of the diiference depends largely on the amount
of dispersion of the variable in proportion to the magnitude of the mean
(cf. Exercise 6.12, p. 150). The geometric mean is necessarily zero, it
should be noticed, if even a single value of X is zero, and it may become
imaginary if negative values occur.

Calculation of the geometric mean

5.30 From equation (5.11) it will be evident that the calculation of
the geometric mean is exactly the same as that of the arithmetic mean
except that instead of adding the values of the variable we add the
logarithms of those values. If there are many values we can draw up
a frequency table for the logarithms and proceed as in Examples 5.1
and 5.2.

Properties of the geometric mean

5.31 The geometric mean is rigidly defined and takes account of all
the observations. It is also fairly easily calculated, though not so easily
as the arithmetic mean. It has, however, no simple and obvious properties
which render its general nature readily comprehensible. This, coupled
with its rather abstract mathematical character, has prevented it from
coming into general use as a representative average.

5.32 At the same time, as the following examples show, the geometric
mean possesses some important properties, and is readily treated
algebraically in certain cases.

{a) If the series of observations X consist of r component series, there
being observations in the first, in the second, and so on, the geo-
metric mean G of the whole series can be readily expressed in terms of
the geometric means Gg, etc., of the component series. For evidently
w^e have at once (as in 5.16 (3)) —

MogG=iVilogGi+iV2logG2+‘ • • • -h-^AogGj. (5.12)

(3) The geometric mean of the ratios of corresponding observations
in two senes is equal to the ratio of their geometric means. For if

X=XJX^

logZ-log Zi-logZ 2

then summing for all pairs of AT^'s and ATg's —

G — Gi/G^ * • • . (5.13)

X = X^X^X^ ... X,

120

THEORY OF' STATISTICS

Xj, X 2 , ... Xrf denoting corresponding observations in r different series,
the geometric mean G of X is expressed in terms of the geometric means
G 2 , . . . Gr by ^be relation

G = GiGgGa . . . Gr . . . . (5.14)

That is to say, the geometric mean of the product is the product of the
geometric means.

5.33 The geometric mean finds applications in several cases where
we have to deal with a quantity whose changes tend to be directly pro-
portional to the quantity itself, e.g. populations ; or where we are dealing
with an average of ratios, as in index-numbers of prices. Suppose,
for instance, we wish to estimate the numbers of a population midway
between two epochs (say two census years) at which the population is
known. If nothing is known concerning the increase 6f the population
save that the numbers recorded at the first census were Pq and at the
second census n years later P„, the most reasonable assumption to make
is that the percentage increase in each year has been the same, so that
the populations in successive years form a geometric series, PqT being
the population a year after the first census, PqT^ two years after the first
census, and so on, so that

== (5.15)

The population midway between the two censuses is therefore

P„/,=Por^/^ = {PoPn)^ .... (5.16)

i.e. the geometric mean of the numbers given by the two censuses. This
result must, however, be used with discretion. The rate of increase of
population is not necessarily, or even usually, constant over any con-
siderable period of time particularly where immigration or emigration are
serious factors.

We shall have more to say about the geometric mean in Chapter 25,
which deals with index-numbers.

The harmonic mean

5.34 The harmonic mean ^^f a series of quantities is the reciprocal of
the arithmetic mean of their reciprocals ; that is, if H be the harmonic
mean,

. (5.17)

The following illustration will serve to show the method of calculation-

AVERAGES

121

Example 5.5. — The table gives the number of litters of mice, in certain
breeding experiments, with given numbers (X) in the litter. (Data from
A. D. Darbishire, Biometrika, 1903, 3, 30.)

Number in
litter

Number of
litters
/

fix

7-000

5-500

5-333

4-250

5-200

-31

5-167

1-571

0-125

0-111

—

121

34.257

Whence

1 34-257

H 121

=0-2831

H==3*532

The arithmetic mean is 4*587, more than a unit greater.

Reciprocal character of arithmetic and harmonic means
5.35 Prices may be stated in two different ways which are reciprocally
related, the resulting arithmetic mean of the one being the harmonic
mean of the other. Supposing we had 100 returns of retail prices of eggs,
50 returns showing six eggs to the shilling, 30 seven to the shilling, and
20 five to the shilling ; then the mean number per shilling would be
6*1, equivalent to a price of 1 *9674. per egg. But if the prices had been
quoted in the form usual for other commodities, we should have had 50
returns showing a price of 2d. per egg, 30 showing a price of l*7i4d. and
20 a price of 2*4d. : arithmetic mean l*994d., a slightly greater value
than the harmonic mean of 1 *967.

The harmonic mean of a series of quantities is always lower than the
geometric mean of the same quantities, and a fortiori, lower than the
arithmetic mean, the amount of difference depending largely on the
magnitude of the dispersion relatively to the magnitude of the mean {cL
Exercise 6.13, p. 150).

SUMMARY

1. Measures of the location or position of a frequency-distribution are
called averages.

122

THEORY OF STATISTICS

2. There are three types of average in general use, the mean (arithmetic,
geometric and harmonic), the median and the mode.

3. The arithmetic mean of N values Xg, . . . Xn is given by

Af=ls(X)

The geometric mean is given by

G=(Zi . . . XN)y^

or IogG=ls(log X)

The harmonic mean is given by

4. The median is the central value of the variable when the values are
ranged in order of magnitude ; if the number of values is even, the median
is conventionally taken to be the arithmetic mean of the two central values.

5. The mode is the value of the variate corresponding to the maximum
of the ideal curve which gives the closest possible fit to the actual distribu-
tion.

6. For distributions of moderate skewness there is an empirical relation-
ship between the mean, the median and the mode expressed by the equation

Mode =Mean — 3(Mean —Median)

EXERCISES

5.1 Verify the following means and medians from the data of Table 4.7,
page 82 —

stature m inches for adult males m
England Scotland Wales Ireland

Mean . . 67-31 68-55 66-62 67-78

Median . . 67-35 68-48 66-56 67-69

In the calculation of the means use the same arbitrary origin as in Example

5.1 and check your work by the method of 5.16 (b).

5.2 The mean of 13 qumbers is 10, and the mean of 42 other numbers is
16. Find the mean of the 55 numbers taken together.

5.3 Find the mean weight of adult males in the United Kingdom from
the data in the last column of Exercise 4.6, page 100. Find the median
weight, and hence find the approximate mode by the relation of 5.27.

AVERAGES

123

5.4 Similarly, find the mean, median and approximate value of the mode
for the distribution of fecundity in race-horses. Table 4.9, page 86.

5.5 Using a graphical method, find the median income subject to sur- or
super-tax in the financial year 1931 from the data of Table 4.5, page 77

5.6 Find the arithmetic mean of the first n natural numbers and show that
it coincides with the median.

5.7 (Data from Agricultural Stattstics, England and Wales, Part 2, 1932.)
The figures in columns 1 and 2 of the small table below show the index-
numbers of prices of certain commodities in the harvest years 1926 and
1931, the years 1911-13 being taken as 100. In column 3 have been added
the ratios of the index-numbers in 1931 to those in 1926, the latter being
taken as 100.

Find the average ratio of prices in 1931 to those in 1926 —

(1) From the arithmetic mean of the ratios in column 3.

(2) From the ratio of the arithmetic means of columns 1 and 2

(3) From the ratio of the geometric means of columns 1 and 2.

(4) From the geometric mean of the ratios of column 3.

Note that, by 5.32, the last two methods must give the same result.

Index-number of

price in

Ratios

Commodity

1926

1931

'31 /’26

(i)

(2)

(3)

Wheat

157

50*3

Fat cattle .

131

118

90*1

Milk

163

139

' 85*3

Eggs

149

110

73*8

Fruit

165

132

80*0

Vegetables

135

158

117 0

5.8 Find the arithmetic and geometric means of the series 1, 2, 4, 8, 16,
. . . 2». Find also the harmonic mean.

5.9 Supposing the frequencies of values 0, 1, 2, ... of a variable to
be given by the terms of the binomial series

• • ■

where find the mean.

5.10 Show that, in finding the arithmetic mean of a set of readings on a
thermometer, it does not matter whether we measure temperature in
Centigrade or Fahrenheit degrees, but that in finding the geometric mean
it does matter.

124

THEORY OF STATISTICS

5.11 (Data from Census of 1901.) The table below shows the population^
of the rural sanitary districts of Essex, the urban sanitary districts (other
than the borough of West Ham), and the borough of West Ham, at the
censuses of 1891 and 1901. Estimate the total population of the county
at a date midway between the two censuses, (1) on the assumption that
the percentage rate of increase was constant for the county as a whole ;
(2) on the assumption that the percentage rate of increase was constant
in each group of districts and the borough of West Ham.

Population

Essex

1891

1901

Rural districts .

232,867

240,776

West Ham

204,903

267,358

Other urban districts

345,604

575,864

Total

783,374

1,083,998

5.12 (Data from Agricultural Statistics, Part 2, 1932.) The following
statement shows the monthly average prices of eggs in England and Wales
in 1932, as compiled from returns from certain markets for National Mark
Specials and English Ordinaries, First Quality, per 120 —

Month

N M Specials

English Ordinaries,
First Quality

s d.

s. d.

January

18 11

15 2

February

15 0

12 11

March .

n n

10 0

April ....

10 10

.9 2

May ....

10 9

8 9

June ....

12 0

10 0

July ....

14 2

12 6

A.uprust

15 6

13 9

September

18 10

16 3

October

20 9

18 9

November

24 1

21 8

December

21 2

16 10

Mean for year '

16 2

13 10

What would have been the mean price for the year in each case if the
wholesale prices had been recorded as retail prices sometimes are, i.e. at
so many eggs per shilling ? State your answer in the form of the equivalent
price per 120, and obtain it in the shortest way by taking the harmonic
mean of the above prices.

CHAPTER SIX

MEASURES OF DISPERSION

Range

6.1 We can now turn to a consideration of measures of the dispersion
of variate values about the central values we have discussed in the last
chapter.

The simplest possible measure of dispersion is the range, i.e. the difference
between the greatest and least values observed. The extreme ease with
which this measure may be calculated and its very obvious interpretation
have led to its use in many industrial problems. There are, however,
objections to the use of the range in fields where speed of calculation
and simplicity of interpretation are not of paramount importance.

In fact, the range is subject to fluctuations of considerable magnitude
from sample to sample. There are seldom real upper or lower limits to
the values which a variable can take, large or small values being only
more or less infrequent. The occurrence of one of these infrequent values
may have quite a disproportionate effect on the range. Suppose, fot
example, we consider the data of Exercise 4.6, page 100 showing the
frequency-distributions of weights of adult males in several parts of the
United Kingdom. In Wales one individual was observed with a weight
of over 280 lb, the next heaviest being under 260 lb. The addition of
this one exceptional man to 737 others has increased the range by some
30 lb, or about 20 per cent.

Moreover, the range takes no account of the form of the distnbution
within the range. We might get the same value for the range from a
symmetrical and a J-shaped frequency-curve. Clearly we could not regard
two such distributions as exhibiting the same dispersion.

6.2 In modern statistics the range finds its chief use in Quality Control,
that is to say, the control of the average quality of a manufactured product.
For instance, when a machine is turning out large numbers of a particular
component, it is customary to examine a smah sample of four or five
taken at, say, half-hourly intervals to see whether the process is remaining
constant within limits of error and is not altering by tool-wear or some
such systematic change. The series of values of mean and range of the
samples can easily be found by comparatively inexpert operators and
are often suflhcient to enable an adequate check to be kept on the
process.

125

126

THEORY OF STATISTICS

6.3 A measure of dispersion should obey conditions similar to those
we laid down for measures of location in the last chapter (5.5). That
is to say, it should be based on all the observations, should be readily
comprehensible, fairly easily calculated, affected as little as possible by
fluctuations of sampling, and amenable to algebraical treatment.

There are three measures of dispersion in general use, the standard
deviation, the mean deviation and the quartile deviation or semi-interquartile
range. We will consider them in that order.

The standard deviation

6.4 The standard deviation is the square root of the arithmetic mean
of the squares of ail deviations, deviations being measured from the arith-
metic mean of the observations. If the standard deviation be denoted by
a, and a deviation from the arithmetic mean by a;, then the standard
deviation is given by the equation

( 6 . 1 )

To square all the deviations may seem at first sight an artificial procedure,
but it must be remembered that it would be useless to take the mere sum
of the deviations, in order to obtain a measure of dispersion, since this sum
is necessarily zero if deviations be taken from the mean. In order to
obtain some quantity that shall vary with the dispersion, it is necessary to
average the deviations by a process that treats them as if they were all of
the same sign, and squaring is the simplest process for eliminating signs
which leads to results of algebraical convenience.

Root-mean-square deviation

6.5 The standard deviation is a particular case of a more general quantity,
known as the root-mean-square deviation, which has theoretical im-
portance.

Let A be any arbitrary value of X, and let ^ (as in 5.11) denote the
deviation of X from A ; i.e. let

Then we may define the root-mean-square deviation s from the origin A
by the equation

( 6 . 2 )

The standard deviation is the value of the root-mean-square deviation
taken from the mean.

6.6 The quantities and s^, i.e, the squares of the standard and root-
mean-square deviations, are sufficiently important in much theoretical
work to have special names.

MEASURES OF DISPERSION

127

The square of the standard deviation, is called the variance.

The quantity i.e. s^, is called the second moment about the

value A, We have already seen (5.11) that the quantity iE(^)is called

the first moment about A, and in the next chapter we shall consider
moments of higher orders.

Thus, the variance is the second moment about the mean.

Relation between standard and root-mean-square deviations

6.7 There is a very simple relation between the standard deviation
and the root-mean-square deviation from any other origin. Let

so that

Then

M-A^d .... ( 6 . 3 )

i ^x+d
= x^^2xd+d^

j^2di:{x) +Nd^

But the sum of the deviations from the mean is zero, therefore the second
term vanishes, and accordingly

Hence the root-mean-square deviation is least when deviations are
measured from the mean, i.e. the standard deviation is the least possible
root-mean-square deviation.

6.8 If a and d are the two sides of a right-angled triangle, s is the
hypotenuse. If, then, MH be the vertical through the mean of a frequency
distribution (fig. 6.1), and MS be set off equal to the standard deviation
(on the same scale by which the variable X is plotted along the base),
S-4 will be the root-mean-square deviation from the point A. This
construction gives a concrete idea of the way in which the root-mean-
square deviation depends on the origin from which deviations are
measured. It will be seen that for small values of d the difference of
s and a will be very minute, since A will lie very nearly on the circle
drawn through M with centre S and radius SM : slight errors in the
mean due to approximationsTn calculation will not, therefore, appreciably
affect the value of the standard deviation.

128

THEORY OF STATISTICS

Calculation of the standard deviation

6,9 If we have, to deal with relatively few, say thirty or forty, ungrouped
observations, the method of calculating the standard deviation is perfectly
straightforward. It is illustrated by the figures below giving the minimum
wage-rates for agricultural labourers in England and Wales at the begin-
ning of 1936.

First of all the mean is ascertained. Then we find the values of x by
subtracting the mean from all values of the variable. Each difference is
squared and the total, obtained. This total divided by the total

frequency is the square of the standard deviation.

In practice, we can simplify the arithmetic by working from an arbitrary
value A instead of from the mean. Such a value is usually known as the

working mean.'' When we have found the mean-square deviation s*
about A we can easily find the value of from equation (6.4).

Example 6.1 — Calculation of Standard Deviation for a short series of
observations (49) ungrouped. Minimum weekly rates’ of wages for
ordinary adult male agricultural workers in England and Wales as at
1st January 1936.

By inspection of the table opposite we see that the mean is in the neigh-
bourhood of 32 shillings. We therefore take this as the working mean A,
The column headed Difference " is the excess of the value of the variable
over this value. The column headed “ (Difference)^ " is the square of
the excess. We find

= -1-612 pence

Hence the mean =32 shillings— 1 -612 pence

= 31 shillings 10*4 pence approximately.

MEASURES OF DISPERSION

129

Area

Wage rates

Difierence

1 (pence)

(Difference)*

Bedford and Huntingdon shires

- 6

Berkshire

-12

144

Bucks

—

Cambridgeshire

- 6

Cheshire

Cornwall

—

Cumberland ....

Derbyshire

2,304

Dorset ....

- 6

Durham

-36

1,296

Essex

-12

144

Gloucester .

-12

144

Hampshire .

-12

144

Hereford

-12

144

Hertford

—

Kent

144

Lancashire (South)

,, (Rest) .

2,916

I-eicester ....

144

Lines (Holland)

576

,, (Kesteven and Lindsey)

-12

144

Middlesex . . ’

400

Monmouth ....

—

Norfolk

- 6

Northants

- 6

Northumberland .

— 6

Notts ....

—

Oxfordshire ....

- 6

Rutland ....

- 6

Shropshire .

—

Somerset ....

Staffs

- 6

Suffolk

-12

144

Surrey ....

Sussex ....

—

Warwickshire

-24

576

Westmorland

-12

144

Wiltshire

-12

144

Worcester

-12

144

Yorks, E Riding

324

,, N Riding

144

,, W. Riding

441

Anglesey and Caernarvon

-12

144

Carmarthen

- 6

Denbigh and Flint

-18

324

Glamorgan

324

Merioneth and Montgomery

-42

1,764

Pembroke and Cardigan

-12

144

Radnor and Brecon

-24

576

Totals

-79

14,539

130

THEORY OF STATISTICS

Also

1 1 4

^ ^2!2y^=.29e • 714 =s2
N ^ 49

a2=rs2~^^2^296-714-(l -612)2

=294-112

0=17-15 pence approximately.

We would direct the student's attention to the necessity for checking
his work at each stage before proceeding to the next. If he neglects this
warning he is likely to learn by bitter experience how essential it was.
For instance, in the above work it w^ould be well to check the value of
the mean by summing the wage rates and dividing by 49. We get in
this way —

Mean = ^-— — ‘ =31s. 10 -4d.

which checks with the mean found from the working mean. Secondly,
the squares of differences should be checked before they are added, and
if the addition is made without a machine, a check should be carried out
by summing first from bottom to top and then from top to bottom, to
avoid repeating errors. A further systematic check is given in 6.11 below.

6.10 If we have to deal with a grouped frequency-distribution the
same artifices and approximations are used as in the calculation of the
mean (5.10 and 5.11). The mid- value of one of the class-intervals is
chosen as the arbitrary origin A from which to measure the deviations g,
the class-interval is treated as a unit throughout the arithmetic, and all
the observations within any one class-interval are treated as if they were
identical with the mid- value of the interval. If, as before, we denote the
frequency in any one interval by /, these / observations contribute to
the sum of the squares of deviations, and we have —

The standard deviation is then calculated from equation (6.4).

6.11 As the arithmetic in calculating the standard deviation is. often
extensive, it is as well' to use some check similar to that of 5.12. In
this case we have —

y (g+l)2^/§2^2/S+/

=S{/|*)+2S{/§)+iv

MEASURES OF DISPERSION

Hence, if we calculate S-j / (g+1)*} as well as 2(/ P), the above equation
gives us a simple check on the accuracy of our work. The following
examples illustrate the method —

Example 6.2 . — Calculation of the standard deviation of stature of male
adults in the British Isles from the figures of Table 4.7, page 82.

(i)

Height

inches

(2)

Frequency

(3)

Deviation
from
value A

(4)

Product

(5)

/(g + 1)

(6)

Product

(7)

/(l+V

57-

-10

- 20

- 18

200

162

58-

- 9

- 36

- 32

324

256

59-

- 8

- 112

- 98

896

686

60-

- 7

- 287

- 246

2,009

1,476

61-

— 6

- 498

- 415

2,988

2,075

62-

169

- 5

- 845

- 676

4,225

2,704

63-

394

- 4

- 1,576

- 1,182

6,304

3,546

64-

669

- 3

- 2,007

- 1,338

6,021

2,676

65-

990

- 2

- 1.980

- 990

3,960

990

66-

1,223

- 1

- 1,223 i

- 4,995

1,223

—

67-

1,329

- 8,584

1,329

—

1,329

68-

1,230

4 - 1

1 1,230

2,460

1 1,230

I 4,920

69-

1,063

-h 2

2,126

3,189

1 4,252

9,567

70-

646

+ 3

1,938

2,584

5,814

10,336

71-

392

+ 4

! 1,568

1,960

! 6,272

1 9,800

72-

202

+ 5

I 1,010

1,212

! 5,050

7.272

73- 1

+ 6

474

553

2,844

3,871

74-

+ 7

224

256

1,568

2,048

75-

+ 8

128

144 1

1,024

1,296

76-

-b 9

50 i

405

500

77-

+ 10

22 1

200

242

Total

8,585

—

8,763

13.759

56,809

65,752

S(/g)= 8,763- 8,584== 179
S j /(g+1)} =13,759-4,995-8,764

This is an example we have already considered when calculating the
mean, and the work of the first four columns is the same as that of Example
5.1, page 107.

As a check on S(/g) we have —

2{/{g+l):-S(/g) =8764-179
= 8585

132

THEORY OF STATISTICS

As a check on 2)(/ we have —

+ “S(/r-)~2i:(/g) = 65,752-56,809-^358

= 8,585
= N

From previous work, M— A =ii' = +0-0209 class-intervals or inches.

56,809
N 8,585

ct2 = 6 *6172 -(0-0209) 2
= 6-6168

a = 2-57 class-intervals or inches.

Example 6.3. — Let us find the mean and standard deviation of the
distribution of Australian marriages given in Table 4.8, page 84.

Calculation of standard deviation of age of bridegroom in a distribution
of Australian marriages.

Age of
bridegroom
(central value)
Years

Frequency

/(§ + !)

16-5

294

- 1,176

882

4,704

2,646

19-5

10,995

-3

- 32,985

- 21.990

98.955

43,980

22*5

61,001

-2

- 122,002

- 61,001

244,004

61,001

25-5

73,054

-1

- 73,054

—

73,054

28-5

56,501

—

56,501

—

56,501

31-5

33,478

66,956

33,478

133,912

34-5

20,569

41,138

61,707

82,276

185,121

37*5

14,281

42,843

57,124

128.529

228,496

40-5

9,320

37,280

46,600

149,120

233,000

43-5

6,236

31,180

37,416

155,900

224.496

46-5

4,770

28,620

33,390

171,720

233,730

49 5

3,620

25,340

28,960 1

177,380

231,680

52-5 !

2,190

17,520

19,710 i

140,160

177,390

55-5

1,655

14,895

16,550

134,055

165,500

58-5

1,100

11,000

12,100

110,000

133,100

61*5

810

8,910

9,720

98.010

116,640

64-5

649

7,788

8,437

93,456

109,681

‘67-5

487

6,331

6,818

82,303

95,452

70-5

326

4,564

4,890

63,896

73,350

73-5

211

3,165

3,376 :

47,475

54,016

76-5

119

1,904

2,023

30,464

34,391

79-5

1,241

1,314

21,097

23,652

82-5

486

513

8,748

9,747

85-5

19'

266

280

5,054

5,600

88*5

100 i

105

2,000

2,205

Total

301,785

—

88,832 i

390,617 i

2,156.838

2,635,287

MEASURES OF DISPERSION

133

We take a working mean ^=28*5.

As a check on S(/^) we have —

^{/(^+1)} -S(/g) =390,617 -88,832
=301,785

As a check on S(/^2) we have —

S{/(g+l)2J -S(/g2)_22(/g) =2,635,287 -2,155,838-177,664

=301,785

Then

M—A =d =0 • 29436 interval

301,785

Hence,

We have —

=0-88308 year
M =29 -383 years

143622 intervals^

301,785

intervals^

=7-056974 intervals^
a =2 -6565 intervals
=7-969, or 8 years approximately.

Sheppard’s correction for grouping

6*12 The student must remember that the treatment of all the values
of a variable in a class-interval as if they were concentrated at the centre
of that interval is an approximation, although, for distributions of sym-
metrical or moderately skew type and class-intervals not greater than
about one-twentieth of the range, the approximation may be a very
close one.

It has been shown that if

(a) the distribution of frequency is continuous, and

(b) the frequency tapers off to zero in both directions,

the variance obtained from grouped data may with advantage be corrected
for the grouping effect by subtracting from it one-twelfth of the square
of the class-interval ; i.e. if the class-interval be h units in width, the
corrected value of the variance and the value obtained from the
grouped data —

(6.5)

134

THEORY OF STATISIICS

The proof of this formula lies outside the scope of this book. We may
emphasise condition (&). The Sheppard correction is not applicable to
J- or U-shaped distributions, or even to the skew form of fig 4.7 {b),
page 84.

Furthermore, unless the total frequency is fairly large, the Sheppard
correction is likely to be of secondary importance compared with fluctua-
tions of sampling (see 19.13). We suggest that, as a general rule, the
correction should not be made unless the frequency is at least 1,000,
or the grouping coarser than that given by intervals of about one-twentieth
of the range. We give in Exercise 6.15 a result which will convey the
general magnitude of the correction for the finer grouping.

Example 6 4. — In Example 6.2 we have —

cji2=6*6168

Here and /i2/12=0-0833

corrected value of — h'^ j\2

=6 -61 68-0 *0833

=6*5335

and a corrected =2 *56, differing from the uncorrected value by 0*01.
Example 6.5. — In Example 6.3 we have —

0-2 (uncorrected) =7 • 056974 intervals^

Here is expressed in terms of and hence to correct it we subtract
iV giving

CT^ (corrected) =6.973641

<7 =2 * 6408 intervals
=7*922 years

as against an uncorrected value of 7*969 years.

Spread of observations and standard deviation

6.13 It is a useful empirical rule to remember that a range of six
times the standard deviation usually includes 99 per cent or more of all
the observations in the case of distributions of the symmetrical or moder-
ately asymmetrical type. Thus m Example 6.2 the standard deviation
is 2*57 in., six times this is 15*42 in., and a range from, say, 60 in. to
75*4 in. includes all but some 36 out of 8,585 individuals, i'e. about
99*6 per cent. This rough rule serves to give a more definite and concrete
meaning to the standard deviation, and also to check arithmetical work
to some extent— sufficiently, that is to say, to guard against very gross
blunders. It must not be expected to hold for short series of observations :

MEASURES OF DISPERSION

135

in Example 6*1, for instance, the actual range is a good deal less than
SIX times the standard deviation.

Properties of the standard deviation

6.14 The standard deviation is the measure of dispersion which it is
most easy to treat by algebraical methods, resembling in this respect
the arithmetic mean amongst measures of position. The majority of
illustrations of its treatment must be postponed to a later stage, but
the work of 6.9 has already served as one example. We showed m 5.16
that if a series of observations of which the mean is M consists of tw^o
component series, of winch the means are and respectively,

and being the numbers of observations in the tw^o component
senes, and iV=A^i +'^2 number in the entire series. Similarly, the
standard deviation cr of the whole series may be expressed in terms of
the standard deviations and Og of the components and their respective
means. Let

Then the mean-square deviations of the component series about the mean
M are, by equation (6.4), and respectively. Therefore,

for the whole senes

. ( 0 . 6 )

If the numbers of observations in the component series be equal and the
means be coincident, we have as a special case —

(6.7)

so that in this case the variance ( 6 . 6 ) of the whole series is the arithmetic
mean of the variances of its components.

It is evident that the form of the relation (6.6) is quite general : if a
series of observations consists of r component series with standard devia-
tions (Jit Og, . . . and means diverging from the general mean of
the whole series by d^, d^, , . , dj., the standard deviation a of the whole
series is given (using m to denote any subscript) by the equation

. . . ( 6 . 8 )

Again, as in 5.16, it is convenient to note, for the checking of arithmetic,
that if the same arbitrary origin be used for the calculation of the standard

136 THEORY OF STATISTICS

deviations in a number of component distributions, we must have —

. . +S(/.g.2) . . (6,9)

6.15 As another useful illustration, let us find the standard deviation
of the first N natural numbers. The mean m this case is evidently
(iV + 1) /2. Further, as is shown in any elementary algebra, the sum of
the squares of the first N natural numbers is

A^(iV+l)(2iV+l)

Applying equation (6.4) we have that the standard deviation a is given

a2=-^(iV+l)(2]V+l)-~J(Ar+l)2

that is,

a2=i(iV2„i) (6.10)

This result is of service if the relative ment of, or the relative intensity
of some character in, the different individuals of a series is recorded not
by means of measurements, e.g. marks awarded on some system of
examination, but merely by means of the respective positions when
ranked in order as regards the character, in the same way as boys are
numbered in a class. With N individuals there are always N ranks, as
they are termed, whatever the character, and the standard deviation is
therefore always that given by equation (6.10).

Another useful result follows at once from equation (6.10), namely, the
standard deviation of a frequency-distribution in which all values of X
within a range ±,1 /2 on either side of the mean are equally frequent,
values outside these limits not occurring, so that the frequency-distribution
may be represented by a rectangle. The base I may be supposed divided
into a very large number iV of equal elements, and the standard deviation
reduces to that of the first N natural numbers when N is made indefinitely
large. The single unit then becomes negligible compared with N, and
consequently

6.16 It will be seen irom the preceding paragraphs that the standard
deviation possesses the majority at least of the properties which are
desirable in a measure of dispersion as in an average (5.5). It is rigidly
defined ; it is based on all the observations made ; it is calculated with
reasonable ease ; it lends itself readily to algebraical treatment ; and we
may add, though the student will have to take the statement on trust

MEASURES OF DISPERSION

137

for the present, that it is, as a rule, the measure least affected by fluctua-
tions of samphng. On the other hand, it may be said that its general
nature is not very readily comprehended, and that the process of squaring
deviations and then taking the square root of the mean seems a little
involved. The student will, however, soon surmount this feeling after a
little practice in the calculation and use of the constant, and will realise,
as he advances further, the advantages that it possesses. Such root-
mean-square quantities, it may be added, frequently occur in other
branches of science. The standard deviation should always be used as
the measure of dispersion, unless there is some very definite reason for
preferring another measure, just as the arithmetic mean should be used
as the measure of position.

Note on nomenclature

6.17 A great deal of confusion has been introduced into statistical
literature by the many different expressions which have been used for
the standard deviation and simple derivatives of it. It used to be almost
a case of tot homines quot nomina, and as the student may meet these
expressions elsewhere, we give a short list of them. The term standard
deviation " is now almost universally accepted, and in th’s book we shall
use no other.

Mean error ” (Gauss), “ mean square error '' and “ error of mean
square '' (Airy) have all been used to denote the standard deviation.

The standard deviation is not to be confused with the standard
error.'' We shall use this term in a special sense, that of the standard
deviation of simple sampling (cf. 17.8),

The standard deviation multiplied by the square root of 2 is also known
as the modulus." The student will see the reason for this multiplication
later. The reciprocal of the modulus is called the " precision."

There is also a quantity known as the "probable error," which is
defined as being 0*67449 times the standard deviation (cf. 17,9). These
last four quantities are particularly important in the theory of errors of
observation and the theory of sampling.

Finally, we may remark that since we shall use the expression
" standard deviation " very frequently, we shall sometimes use the
abbreviation " s.d." or simply the symbol a.

Mean deviation

6.18 We have already remarked that it would be useless to take the
sum of deviations from the mean as a measure of dispersion because such
sum is identically zero. We therefore remove the signs of the deviations
by squaring to reach the standard deviation.

It is also possible to overcome this difficulty by adding the sum of
deviations taken regardless of sign. The arithmetic mean of these
" absolute " deviations is called the mean deviation.

138

THEORY OF STATISTICS

If we write j ^ j to denote the deviation from an arbitrary value A taken
as positive whatever its actual sign, the mean deviation is thus defined as

m.d.=is(i^i) (6.12)

(The expression |g| is read “modg — an abbreviation for ''the modulus

of r')/

6.19 Just as the root-mean-square deviation is least when deviations
are measured from the arithmetic mean, so the mean deviation is least
when deviations are measured from the median. For suppose that, for
,some origin exceeded by m values out of iV, the mean deviation has a value
Ae Let the origin be displaced by an amount c until it is just exceeded by
m—1 of the values only, i.e. until it coincides with the mth value from the
upper end of the series. By this displacement of the origin the sum of
deviations in excess of the origin is reduced by me, while the sum of
deviations in defect of the mean is increased by (N —m)c. The new mean
deviation is therefore

A +

{N — m)c— me
N

=A+i(iV-2w)c

The new mean deviation is accordingly less than the old so long as

That IS to say, if N be even, the mean deviation is constant for all
origins within the range between the N /2th and the {N /2+l)th observa-
tions, and this value is the least ; if iV be odd, the mean deviation is lowest
when the origin coincides with the (iV’+l)/2th observation. The mean
deviation is therefore a minimum when deviations are measured from the
median or, if the latter be indeterminate, from an origin within the range
in which it lies.

Calculation of the mean deviation

6,20 The mean deviation is perhaps most easily calculated about the
mean, which is always determinate, except in the case of distributions with
an indeterminate final class. As, however, it is a minimum about the
median, we sometimes require to know the value about that point. The
following examples will make the method of calculation clear.

MEASURES OF DISPERSION

139

Example 6.6. — Let us find the mean deviation about the mean and
about the median in the ungrouped data of Example 6.1.

The data were arranged in alphabetical order of the county wage areas,
which makes it a little difficult to ascertain the median by inspection. On
rearranging in order of magnitude, we find that the median is the value
3is. 6d.

The deviations from the median value are, then, in order of magnitude

-~36, -30, -18, -18, -12, -6 (12 times), 0 (10 times),

6 (7 times), 9, 12, 12, 12, 15, 18, 18, 18, 24, 24, 26, 27,

30, 54, 60

The sum of the negative deviations =—186
The sum of the positive deviations = 401
Hence the sum of absolute deviations = 587

587

Hence m.a.=-^=12 pence approximately.

To find the m.d. about the mean, 31s. 10* 4d., we note that the 27
negative or zero deviations from the median would be increased by 4-4
pence on transferring to the mean, and the 22 positive deviations decreased
by 4*4 pence. The net effect on the total absolute deviations is then an
increase of (27— 22) x 4*4 pence =22 pence.

Hence the m.d. about the mean is —

^ 22
49^^49

=12*43 pence

Example 6.7. — Let us find the mean deviation of heights about the
mean in the data of Example 6.2.

In the case of a grouped frequency-distribution the sum of deviations
should first be calculated from the centre of the class-interval in which the
mean (or median) lies and then reduced to the mean (or median) as
origin.

In this case the mean lies in the interval 67-. We found when calculat-
ing it that the negative deviations totalled —8584 and the positive devia-
tions 8763. Hence the sum of absolute deviations from the centre of the
interval is 17,347 — the unit of measurement being the class-interval.

To reduce to the mean as origin we note that if the number of observa-
tions below the mean is Ni and above the mean and M—A=d as
before, we have to add N^d to the sum when found and subtract N^d, In
this case d===Q*02 class-interval, ^^=4,918 and iN72=3,667.

140

THEORY OF STATISTICS

Hence we must add

(4,918->3,667)x0*2=-f25 intervals

i.e. the total of deviations = 17,372

and

17 37^^

m.d. = - ^ ^ =2'02 intervals or inches.

8,585

The mean deviation from the median should be found in a similar way,
the calculation being assisted if the class-interval in which the median
lies is taken as origin,

6,21 As in the case of the standard deviation, the above calculations
assume for certain purposes that all the values of the variable can be
treated as if they were concentrated at the centres of class-intervals. This
gives sufficient accuracy for all practical purposes if the class-intervals are
reasonably narrow. It has not been found possible to give any simple
correction, such as Sheppard's correction, for errors of grouping in the
mean deviation, but we give at the end of this chapter an Exercise (6.11) as
to the correction to be apphed if the values in each interval are treated
as if they were evenly distributed over the interval instead of being
concentrated at its centre.

Empirical relation between mean and standard deviations for symmetrical
or moderately skew distributions

6.22 It is a useful rule for the student to remember that for symmetrical
or moderately skew distributions the mean deviation is about four-fifths
of the standard deviation. Thus, for the distribution of male statures
of Examples 6.2 and 6.7, we have —

m.d.

s.d.

2-02

2-57

=0*79

For the short series of observations of Example 6.1 —

m.d.

s.d.

12'43

17-15

=0-72

Quartiles

6.23 A natural extension of the idea of the median consists in ascer-
taining the variate values Qi and such that one-quarter of the observa-
tions lies below and one-quarter above In this case clearly one-
quarter lies between and Mi, the median, and one-quarter between Mi
and

is termed the lower qmrtile and the upper quartile. The quartiles
and the median thus divide the observed values of the variable into
four classes of equal frequency.

MEASURES OF DISPERSION

141

We saw that if the number of observations was even, there was an
indeterminacy in the position of the median which required the additional
convention that in such cases the median would be taken to be mid-way
betw^een the two central values. Similar indeterniinacies may arise in
hxing the quartiles unless the number of observations is one less than a
multiple of four. Such cases are treated in an analogous way by supple-
mentary conventions, which will be clear from the following examples.

Example 6,8. — To determine the qaarliies of the data of Example 6.1.

Here there are 49 observations, and so the 25th gives the median.
We regard half the 25th observation as failing below’ the median and half
above. The lower quartile must divide into two equal parts the 2A\
observations falling below the median. The observations other than the
median are —

28/6, 29/-, 30/-, 30/-, 30/6, 31/- (12 times), 31 /6 (7 times).

The lower quartile must divide the 24| observations into two sets of
12J. The 12th and the 13th values are both, as it happens, 31 /-, and
being between the two is thus 31 /• also.

The 24 observations between the median and the highest value are —

31 /6 (twice), 32 /- (7 times), 32/3, 32/6 (3 times), 32/9, 33 /- (3 times),
33 /6, 33 /6, 33 /8, 33 /9, 34 /-, 36 /-, 36 /6.

The i2th and 13th observations are both 32/6, and herxe this is the
value of Q^.

If the 12th and IStli observations had been, say, 32/6 and 33/-. we
might have taken to be 32/6 but regarded \ of the 12th observation
as l^dng above that value.

Example 6.9, — To determine the quartiles of the distribution of Example

6 . 2 .

Data of this kind are tieated by simple arithmetical interpolation or
graphical interpolation on the lines of 5.20 or 5,21,

The quartiles are to divide the distribution into four equal parts. We
have, therefore

8585

“4“

=2146 ‘25

To the interval 65- are 1,376 individuals

Difference =770 ' 25

770 * 25

Hence, is - — in. from the beginning of the interval, which is 64

Ci=65-71

Similarly, from the interval 70- onwards are 1,374 individuals.
Diflerence from 2146 -25 =772 -25.

142

THEORY OF STATISTICS

Hence,

<33=69f

772*25

1063

=69*21 inches

It IS left to the student to check the values by graphical interpolation.

Quartile deviation

6.24 If Mi be the value of the median, in a symmetrical distribution

and the difference may be taken as a measure of dispersion. But as no
distribution is rigidly symmetrical it is usual to take as the measure

q_ Qz~‘Qi
^ 2

and Q is termed the quartile deviation, or better, the semi-interquartile
range — it is not a measure of the deviation from any particular average.
Thus, from the values calculated in Example 6.8 we have —

3 2/6-31/-
^ 2

LVJU.. -

=9 pence

and from Example 6.9 we have —

69*21

-65*71

1*75 inches

Empirical relation between quartile and standard deviations
6.25 For symmetrical and moderately skew distributions the semi-
interquartile range is usually about two-thirds of the standard deviation.
Thus, for the height distribution of Examples 6.2 and 6.9,

1-75

’2-57

0-68

For the wage statistics of Examples 6.1 and 6.8,

IT'-IS

=0-52

which is considerably lower. We should, however, hardly have expected
the comparatively few observations comprised in these data to conform at
all closely to the empirical relation.

MEASURES OF DISPERSION

143

6.26 It follows from this relation that a range of 6 times the standard
deviation corresponds to a range of 9 times the semi-interquartile range
(and 7*5 times the mean deviation). Within these ranges we expect to
find at least 99 per cent of the observations in symmetrical or moderately
skew distributions.

Comparison of the three measiires of aispersion

6.27 The semi-interquartile range has two advantages over the standard
deviation and the mean deviation ; it is calculated with great ease, and
it has a clear and simple meaning.

In almost all other respects the advantage lies with the standard
deviation. The semi-interquartile range has no simple algebraical pro-
perties, and its behaviour under fluctuations of sampling is difficult to
decide. In ail but the most elementary statistical work these are over-
whelming disadvantages, and the use of the semi-inter quaitiie range is not
to be recommended unless the calculation of the standard deviation has
been rendered difficult or impossible, e.g. owing to the employment of
irregular class-frequencies or of an indefinite terminal class.

Absolute measures of dispersion

6.28 The three measures of dispersion we have been discussing have
ail been expressed in terms of the units of the variate ; e.g. the standard
deviation of height-frequencies was found in inches, and the mean deviation
of wage-frequencies in pence. It is thus impossible to compare dispersions
in different populations unless they happen to be measured in the same
units.

For this reason some, statisticians have recommended the use of
absolute measures of dispersion, which shall be pure numbers and
not expressible in some particular scale of units. Such measures would
permit of comparison between populations of very different natures.

It is easy to construct several coefficients of the kind required. The
standard deviation and the mean deviation have the dimensions of the
variate, and it is only necessary to divide them by another factor which
has the same dimensions ; e.g.

Mean deviation Mean deviation Standard deviation

Mean ’ Mode Mean

are ail of the required type.

Coefficient of variation

6.29 The last-mentioned in the foregoing paragraph in a modified
form is the only coefficient which has come into general use. We define
the coefficient of variation, v, as

. (6.13)

144

THEORY OF STATISTICS

This coefficient is obviously rather unreliable if the mean is near to
zero ; but provided the nature of the ratio is kept in mind the coefficient
may be useful in comparing the variation of materials which emanate
from populations of the same type.

Reduction of frequency-distribution to absolute scale

6.30 Comparability of form mav, however, be reached in a different
way ; that is to say, by regarding a itself as a unit and expressing other
measures in terms of it. Thus, in the height distribution of Example
6.2, 0- = 2*57 inches, or 1 inch =0*389 a. Hence the intervals are 0*389 a
in width, and run: 57x0*389 a- 58x0*389 a-, etc. ; i.e. 22*173 a~,
22*562 a-, etc.

A distribution expressed in this way has unit standard deviation, for

The distribution reduced to the scale of a may thus be regarded as
expressed in absolute '' units, and two distributions expressed in this way
may readily be compared as regards form, but not as regards dispersion,
for this has been made the same in the two cases.

Deciles and percentiles

6.31 We may conclude this chapter by describing briefly methods
which have been much used in the past in lieu of the methods described
in this and the preceding chapter.

Instead of dividing the total frequency into 4 parts by quartiles, we
may divide it into 100 parts by what are called percentiles. Or we may
divide into 10 parts by deciles. The theory of these quantities is precisely
analogous to that of the quartiles : there may, for instance, be certain
indeterminacies in their exact definition which are removed by supple-
mentary conventions ; they can be obtained by arithmetical or graphical
interpolation ; and they have simple and obvious meanings.

Quantities such as quartiles, deciles, etc., which divide the total fre-
quency into a number of parts, are called quantiles or grades, and when we
speak of the grade of an individual we mean thereby the proportion of the
total frequency v/hich lies below it. Conventionally, half the individual
IS regarded as lying above, and half belo^r. the point determined by the
variate value which it bears..

The distribution curve

6.32 The grades or quantiles may conveniently be found by a graphical
method which is an extension of that of 5,21 . Against the variate- value
as abscissa we graph as ordinate the cumulated frequency up to and in-
cluding the corresponding variate-value. This is called the distribution
curve. By reading off the ordinate corresponding to a given variate we

MEASURES OF DISPERSION

X45

can find, approximately at least, the number of members of the population
bearing that or a lower value. Similarly, by reading of the variate
corresponding to a given ordinate we can find the quartiles, just as we
found the median in 5.21. In figure 6.2 we show the distribution curve
for the data of Example 6.2, with the lines corresponding to the median
and the quartiles. Figure 5.3 is really an enlarged version of part of this
curve.

A somewhat similar form of graph (with the percentiles as abscissa and
the variate as ordinate) was formerly in use and was known as Galton's
ogive. The curve was not, however, always shaped like an ogive. The
distribution curve appears to provide a more natural method of representa-
tion and a better name. The mathematical reader will recognise it as
the graph of the integral of the frequency curve.

6.33 An extension of the method of quantiles to the treatment of non-
measurabie characters has also become of some importance. For example,
the capacity of the different boys in a class as regards some school subject
cannot be directly measured, but it may not be very difficult for the
master to arrange them in order of merit as regards this character : if the
boys are then “ numbered up ” in order, the number of each boy, or his

Height (inches)

Fig. 6.2. — ^Distdbation cturve for stature
(Same data as fig. 4.6, p. 83)

146

THEORY OF STATISTICS

ranki, serves as some sort of index to his capacity. It should be noted
that rank in this sense is not quite the same as grade ; if a boy is tenth,
say, from the bottom in a class of a hundred his grade is 9*5, but the
method is in principle the same as that of grades or quantiles. The
method of ranks, grades or quantiles in such a case may be a very serviceable
auxiliary, though, of course, it is better if possible to obtain a numerical
measure. But if, in the case of a measurable character, the quantiles
are used not merely as constants illustrative of certain aspects of the
frequency-distribution, but entirely to replace the table giving the
frequency-distribution, serious inconvenience may be caused, as the
application of other methods to the data is barred. Given the table
showing the frequency-distribution, the reader can calculate not only
the quantiles, but any form of average or measure of dispersion that has
yet been proposed, to a sufficiently high degree of approximation. But
given only certain quantiles such as the percentiles, or at least so few of
them as the nine deciles, he cannot pass back to the frequency-distribution,
and thence to other constants, with any degree of accuracy. In all cases
of published work, therefore, the figures of the frequency-distribution
should be given ; they are absolutely fundamental.

Gini’s mean difference

6.34 The Italian statistician Corrado Gini has proposed a measure of
dispersion which at first sight seems to have certain advantages over the
standard deviation. It is the mean of the differences (taken regardless
of sign) of each possible pair of variate values exhibited by the population ;
e.g., if the frequency of the value is //, the coefficient of mean difference is

^ ^ • • • (6-14)

or, if we regard each member as taken with itself, contributing nothing
to the sum in (6.14) but increasing the number of pairs of values to
instead of iV’(iV— 1), we have the coefficient of mean difference with
repetition —

££ . . . . (6.15)

6,35 These coefficients are more difficult to calculate than the standard
de%iation or the mean deviation, but they have a theoretical attraction
in that they depend on the differences of values between themselves and
not on the spread about some arbitrary point such as the mean or the
median. They thus measure, in a sense, the intrinsic spread of the
population independently of an origin of location.

MEASURES OF DISPERSION

147

A similar property, however, is possessed by the standard deviation.
Suppose that, in equation (6.15), is’e sought to obviate the difficulties
of using absolute values by defining a new coefficient E by the similar
expression.

• ■ (6.16)

Since {x^ —Xk) ^ +Xk^ —2xjXk

and

s £ 1 ( /^ /* ) = f 2 /, ) ^ 2 /. )

we find

£2=^^^ |iV2s2+iV2s2-2iVS^i2|

=2{s^-d^)

=2ct*

(6.17)

SO that E is merely the standard deviation multiplied by \/2. This relation
shows that, apart from the constant -^^2, the standard deviation may be
regarded as the root-mean-square of all possible pairs of differences of
the variate values. Such being the case, the mean difference of Gini
loses most of its relative theoretical attraction, and as it is more di^cult
to calculate the balance of advantage remains with the standard deviation.

SUMMARY

1. The standard deviation o is defined by

a*=l2(*^)

where x is the deviation from the arithmetic mean, is called the
** variance.”

2. The root-mean-square deviation s about a point A is defined by

s*=i2(g*)

where | is the deviation from A.

148

TKEOKY OF STATISTICS

3. If M —A then

s2=cr^+^^

4. For grouped data the variance should be corrected by subtracting
~ , where k is the width of the class-interval, provided that {a} the

frequency is continuous, and (b) that it tapers off to zero in both directions.

5. The s.d. is the minimum root-mean-sqiiare deviation.

6. The mean deviation is defined as

m.d.=is{ K 1 ).

7. The m.d. is a minimum about the median.

8. The quartiles are the values of the variate which divide the total
frequency into 4 equal parts ; similarly, the deciles divide it into 10 equal
parts and the percentiles into 100 equal parts.

9. The quartile deviation, or semi-interquartile range, is deSned as

^ ' 2

10. For symmetncal or moderately skew distributions,

m.d. =0* 8a and approximately.

11. For the majority of such distributions 99 per cent of the total
frequency lies within a range of 6a, 7-5 m.d. or 9Q.

EXERCISES

6.1 Verify the following for the data of Table 4.7, page 82 (in continua-
tion of the work of Exercise 5.1) —

Standard deviation (uncorrected)

Mean deviation ....

Quartile deviation ....
Mean deviation /standard deviation .
Quartile deviation /standard deviation
Lower quartile. . * .

Upper ,,

Stature in inches for adult males born in

England

Scotland

Wales

Ireland

2*56

2-50

2-35

2-17

2-05

1-95

1-82

1-69

1-78

1-56

1-46

1-35

0-80

0-78

0*69

0-62

65-55

66-92

65*06

66-39

69-10

70-04

67-98

69-10

MEASURES OF DISPERSION

6.2 Find the standard deviation, mean deviation, quartiles and serni-
interquartile range for the data in the last column of the table of Exercise
4.6, page 100 (in continuation of the work of Exercise 5.3).

Compare the ratios of mean and qiiartile deviations to the standard
deviation with those stated in 6,22 and 6.25 to be usual for moderately
skew distributions.

6.3 Using, or extending if necessary, your diagram for Exercise 5.5,
page 123, find the median and upper quartile for incomes subject to sur-
er super-tax.

Find also the 9th decile (the value exceeded by 10 per cent of incomes
only).

6.4 Find the quartiles of the distribution of Australian marriages given
in Example 6.3, and find the semi-interquartile range.

6.5 Find directly the standard deviation of the natural numbers from
1 to 10 , and hence verify equation ( 6 . 10 ).

6.6 Show that, for any distribution, the standard deviation is not less
than the mean deviation about the mean.

6.7 Show that, for a J-shaped distribution with the maximum frequency
towards the lower values of the variate, the median is nearer to Qi than
to § 3 .

6 . 8 , Find the mean and standard deviation of the following numbers
( 1 ) without further grouping, (2) grouping the numbers by fives (40-, 4S~,
50~, etc.), (3) grouping by tens (40-, 50-, etc.) —

40, 43, 43, 46, 46, 46, 54, 56, 59, 62, 64, 64, 66 , 66 , 67, 67, 68 , 68 ,

69, 69, 69, 71, 75, 75, 76, 76, 78, 80, 82, 82 82. 82, 82, 83, 84.

86 , 88 , 90, 90, 91, 91. 92, 95, 102, 127.

6.9 Apply Sheppard's correction to the standard deviations calculated
in Exercises 6.1 and 6,2 above.

6.10 (Continuing Exercise 5.9, p. 123.) Supposing the frequencies of
values 0 , 1 , 2 , 3, . . . of a variable to be given by the terms of the binomial
series.

n(n—l)

q», nq'-^. . . .

where find the standard deviation.

6.11 (Cf. the remarks at the end of 6.21.) The sum of the deviations
(without re^rd to sig:n) .about the centre of the class-interval contaiaing
the mean (or median), in a grouped frequency-distribution, is found to be
S. Find the correction to be applied to this sum, in order to reduca ;t
to the mean (or median) as origin, on the assumption that the observations

150

THEORY OF STATISTICS

are evenly distributed over each class-interval. Take the number of
observations below the interval containing the mean (or median) to be
in that interval Wg above it n^, and the distance of the mean (or
median) from the arbitrary origin to be d,

0J2 Show that if deviations are small compared with the mean, so that
(xjM)^ and higher powers of x/M may be neglected, we have approxi-
mately the relation

where G is the geometric mean, M the arithmetic mean and a the standard
deviation : and consequently to the same degree of approximation

6. ^ 3 Similarly, show that if deviations are small compared with the mean,
W’e have approximately

E being the harmonic mean.

6.14 Find the coefficients of variation of the height distributions of
Exercise 6.1 (using the uncorrected values of the s.d. as given).

6.15 Show that if a range of six times the standard deviation covers at
least 18 class-intervals, Sheppard's correction will make a difference of
less than 0*5 per cent in the uncorrected value of the standard deviation.

CHAPTER SEVEN

MOMENTS AND MEASURES OF SKEWNESS
AND KURTOSIS

MoMesats

In considering the calculation of tne mean and the rooi-mean-

square deviation we have defined, in passing, the quantities— I (/^) and

as the first and second moments about the value A, ^ being as

before the value X —A, i.e. the excess of the variate value X over the value
A. The first moment about the mean is zero, and the second moment
about the mean is the variance (6.6) .

In generalisation of these definitions we now define the nth. moment
about A as /in, where

/u'=ls(/a (7.1)

The moments about the mean, which are of particular importance,
we write without dashes so that

... . . ( 7 . 2 )

From these definitions we have —

i“o'=/'o=4^(/)=l g® and

H =0

11%

These results we have already seen.

151

152

THEORY OF STATISTICS

7.2 The word moment derives from Statics, and we may direct
the attention of the student is familiar with moments of forces to the
fact that the sum is divided by N in the definition above. This

amounts to a slight departure from the Statical practice, and some writers
refer to what we have called "moments as '' moment-coefiicients '' in
order to keep this fact in mind. In Statistics, however, no confusion is
likely to arise from the use of the briefer form " moments/'

Moments about the mean in terms of moments about any point
7.3 We have, by definition,

:==={X--M)+{M-A)

Hence,

and

=xA-d

Now, by the binomial theorem,

. . . +d^

Hence,

Dividing by N we get —

• • (7-3)

Similarly,

and

. . +(-I)*i» . (7.4)

Tkese useful relations express the moments about the mean in terms
of those about an arbitrary point A, and vice versa,

Irs particular we have —

If n—l,

tiJ from (7.3)

H from (7.4)

which are sknp'y the relation M - J in another form.

If w=2,

from (7.3)

-f

P'% =i®s'— froni (7.4)
==li/-2d^+d^

These are the relation d^.

MOMENTS, SKEWNESS AND KURTOSIS

153

If w=3,

from (7.3)

• ' • • • (7*5)

/^3 ==/^3'— from (7.4)

=/i^'~~3dju,'+2d^ .... ( 7 . 6 )

= 11 ^ + 4 ^ 6/43 +6^^ from (7.3)

• • • (7.7)

+6i V 2 " from (7.4)
=/44'~4i^/43'+^^^V2"~‘3ii^ • • • f^-S)

Calculation of moments

7.4 The calculation of moments of the third and higher orders is similar
to that of the first and second. For grouped data we regard the observa-
tions as concentrated at the mid-points of the intervals ; we choose a
convenient arbitrary origin A, find the moments about it and use the
relations (7.3) and (7.4) above to find the moments about the mean ; we
use a check on the arithmetic similar to that of 6.11 ; and we have under
certain conditions. certain Sheppard corrections for grouping.

In practice we rarely require to ascertain moments higher than the
fourth. Indeed, moments of higher orders, though important in theory,
are so extremely sensitive to sampling fluctuations that values calculated
for moderate numbers of observations are quite unreliable and hardly ever
repay the labour of computation.

7.5 There are various checks in use for the arithmetic of calculation.
We shall use a generalisation of the simple identities of 5.12 and 6.11
In fact, we have

(g + l)3==:g3+3p+3^^_l

and hence,

s { /(l+l)-’] =S(/a +3S(/g^)-f 32:(/|) +iV

Similarly,

2 l/(g+l)^} ==2(/a+4S(/a+6E(/a+4S(/g)+i^
and so on.

Thus, in calculating we also find this,

together with the sums of lower orders, will give us a ready check on the
work.

Example 7.1. — Continuing our work on the height distribution of
Table 4.7, page 82, let us find the third and fourth moments of the
distribution about the meaxi.

In almost all practical v/ork we require the first and second moments
as a matter of course. It is therefore best to proceed systematically in

THEORY OF STATISTICS

O'.^^rJ^CD<NC0X’^

cscAcn'^^cDOOr--*co

a> occxoe^iaiN^oc^
O? XOt^QOit^t^lNOX
X X x 0^0) o^C'1^

C> <0 litT ui r-T Ci r~r V o* oT

r-t X <x> Th-eo X x o ir> (M

r-< Cl C^ ^ r-< 1 -.

OXXCIO'^iNXiftO

xo<Nu:>mxxcooo

d O X X CM X t/s o^

r-*" K d o xT d' cc ic d o

r-HlOOdOl>-XXd

xxdxxxxdo

X'<S‘OXtNir-^COX©

’^OXXXXXCO©

CO^

Oi-''Tt«Odt>.Ti<'>3r'Od

•^0'^OX05XCOOO

o6 i> co^ co^ co^ q^ q

— ' d' CO o' o' o" lei

1 1 1 i 1 1 1 ) 1

lii

cxT ao t-T 05“ CO in cd y^ x' d"

d Tj* '«f d t-i i-<

ocoxxxxcowo
o^cocoddT-'cpd
0^05 ^^O 05 i-H d^o 05

C0_^

OTt‘dxO'^codxo
eCO'<t50XC0C>05^O
d^x o d o 05 ^^qo

d*^ d‘ K •^'' t> r-T ui aS s>

r-< 1-t d d T-t

M (1 1 1 1 i i

y^ CO in' x" x' r^' o' CO CO d‘

^ d d ^

d<ococox'«j‘cocoo

C0XXC«.r>*OrM>05

*-dCO'?POt>XC005

OINCOOd»-iXCOOd

dcocoor^t^'^a50'ri<

<qx CO x^d X o d^x d

d" d' CO oi

r-T

©* o" Oi pc CO d'

0^X<j5XlftTt*^0
Od^QOOdOdX
d X X O 05 d CO 0^05
cf d" '4'“ X X x*

Od'^d©'^XTj‘XO
«ic»-<tNtn:g?©dOo
d d^X d^O^X W5 d
tT W5 U5 d" y^

XdXCOiCXdXO
d 1-* X ©

I I M M I I I

y-*

OOi^OdXX-^Od
XXXX»^»nW5*?Mi3d
-^i-txaSduodi-^
d* cr: d*

oxdc^xxxt:?©

dC0'-^X05-;sei>OX
f^d-^ 00X005

OOXXO'^’^XXO

Xd«c5^iN.ddTt*d

df«<05XO'<tdrH

-d*'

d"«S*’^»-^«05‘^050
r-t Tp X CO 05 CP 05

CO 5o CO t>. c^ i> CN r> h* i>

Total 8,585 — • | 8,763 13,759 56,809 65.752 | 119.391 1 236,653 1,182 061 | 1,539,292

MOMENTS, SKEWNESS AND KURTOSIS

155

the computation of the vsirious moments by setting out the arithmetic in
tabular form as on opposite page.

From this table we have —

^ifi) = 8,763 - 8,584= 179

S(/|") = 56,809

=119,391 -117,622= 1,769

=1,182,061

As a check on Z( /■ P) we have —

5:(/|®)+32(/a+3S(/|)+iV
=1,769+170,427 +537 +8,585
=181,318

=s{ f(^+ir}

As a check on 2( f P) we have —

2( f^*) +4S(/|S) +6S(/|2) +4S(/g) +Ar
=1 .182,061 +7,076 +340,854 +716+8.585
= 1,539,292
=S{/(?+l)*}

We have then-

^=Ai'=^S(/g)=^= 0-020,850,32

56,809

6-617,239,37

=137-689,108,91

" ‘ 8,585

^ = 0-206,057,08

=137-689,108,91

0,OoO

M'i

=6-616,805

From equation (7.6) —

=0 - 206,057,08 -0 • 41 3,914,67 +0 - 000,01 8, 1 3
=-0-207,839

From equation (7.8)—

— 4«f/is' +6^i Vs' — 3if*

=137-689,108,91 -0-017,184,24 +0-017,260,51 -0-000,000,57
=137-689,185

which gives us /ij, /ij, ^4 in units based on class-intervals, i.e. inches.

Totals 301,785 — 318,049 474,490 2.155,838 2.635,287 j 13,675.105 19.99:

MOMENTS, SKEWNESS AND KURTOSIS X57

Example 7.2.— To find the moments about the mean of the distribution
of Australian marriages of Table 4.8, page 84,

Until the last stage we work in ciass-intervals of 3 years. As in Example
6.3, page 132, we take a working mean at 28*5 years.

From this table we have —

S(/g) = 318,049 -229,217

S(/g«)=13, 675, 105 -876, 743

s(/a

As a check on S(/g) we have —

2( / g) 4-^=88,832 4-301,785 =390,617

=S{/(^4-l)}

Similarly, for S(/|*) —

S(/|®)4-2S(/^) 4- A^=2, 155,838 4-177, 664 4-301.785
=2,635,287
=S{/(f4-l)“}

= 88,832

= 2,155,838
= 12,798,362
-=137,306,162

As a check on S(/|®) —

S(/|»)4-32(/g*)-h3S(/g)4-2V

=12,798,3624-6,467,5144-266,4964-301,785

=19,834,157

=S{/(g4-l)®}

As a check on S(/ —

S(/g*) 4-42(/^«) 4-6S(/a 4-4S(/0 4-iV
=137,306,1624-51,193,4484-12,935.0284-355,3284-301,785
=202,091,751
=S{/(|-M)«>

Hence, about the working mean-

a:=/t/=

h'=

88,832

301,785

2,155,838

301,785

12,798,362

301,785

137,306,162

301,785

= 0-294,355,253
= 7-143,622,115
= 42-408,873,867
=454-980,075,219

158

THEORY OF STATISTICS

For moments about the mean —

/^2 =A2' =7 * 056,977

/^3 -M 3 ' 2 ' =36*151, 595

+6iV2' --3^^ =408 • 738,210

These are expressed in class-intervals, which are hnits of three years.
If, as we rarely do, we wish to express the results in other units, say one
year, we must multiply the first moment by 3, the second by 3^, the third
by 3^, the fourth by 3^, and so on ; e.g.

/ig =7 • 056,977 X 9 =63 • 5 12,79

In tliis and the preceding example we have retained more digits than
are probably necessary, but the student will find it as well to retain several
more than appear to be required, since subsequent work involving multi-
plication or addition may otherwise throw doubt on the final figures.

7.6 It will be evident that the labour involved in calculating the third
and fourth moments is very considerable. Calculating machines or
tables of powers are a great help, and certain tables for the specific purpose
of computing moments will be found in Tables for Statisticians and
Btomeiricians, Part The student should familiarise liimself with the
methods given in the two examples above, since, although we shall not
use them to any great extent in this book, moments are important in
more advanced theory.

Sheppard corrections for moments

7.7 As in the case of the second moment, the effect due to grouping
at mid-points of intervals may be corrected for by formulae due to W. F.
Sheppard, from whom they derive their name. The formulae for the
second, third and fourth moments are as follows —

/tg (corrected) =/4a---
(corrected) = 7^3

(corrected) =iia~Wh

(7.9)

where h is the width of the class-interval. If we are working in class-
intervals as units, h is taken to be unity.

The use of these formulae is restricted to the cases which we mentioned
in 6.12, i.e. those in which (a) the frequency-distribution is continuous,
and (fc) the distribution tapers off to zero in both directions.

MOMENTS, SKEWNESS AND KURTOSIS

159

Example 7.3. — In Example 7.1 we found —

/i2= 6*616,805
^43 = ^0*207,839
;^4 = 137*689,185

Applying the above corrections, h being 1 —

//g (corr.)== 6*616,805 -0*083,333
= 6*533,472
//g (coiT.) = — 0*207,839

(corr.) =137* 689, 185 -3 * 308,402 +0 * 029, 1 67
= 134*409 950

Example 7.4. — In Example 7.2 we have, in units of 3 years —

7*056,977
^3= 36*151,595
=408 *738,21

Thus —

fcorr.)= 7*056,977-0*083,333
= 6*973,644
/^3 (corr)= 36*151,595

fi , (corr. ) =408 • 738,2 1 0 -3 * 528,489 +0 • 029, 1 67
=405*238,888

In units of one year the corrected moments are given by multiplying
by 9, 27 and 81 as before.

and y-coeffkients

7.8 Certain quantities calculated from the moments about the mean
are of particular importance in statistical work. We define —

and two further quantities —

. (7,10)
- (7.11)

ri=+VA (7.12)

. . - (7.13)

The reason for the introduction of these arbitrarydooking quantities will
appear in the sequel.^

In general, Karl Pearson defined

IHn {-i

i6o

THEORY OF STATISTICS

It IS to be noted that these four coefficients are aU pure numbers and,
as such, are independent of the scale of measurement of the variable ; for
since fin has the dimensions of (variable)", fi^^ has the dimensions (variable)®
and so has fi^, and hence their quotient has dimension zero, i.e. is a pure
number ; and similarly for the quotient of fi^ and fi^^.

Example 7.5. — Let us calculate and for the distribution of Example

7.1.

We have, using the corrected values of Example 7.3 —

_(-0-207839)®

(6-533472)®
0-043197

278-889

0-000155

_134 -40995
~ 42-68662
=3-149

Example 7.6. — Similarly, in the data of Example 7.2, using corrected
values —

_(36-151595)®

(6-973644)»

=3 -854
_405- 238888
(6 '973644) 2
=8*333

It should be noted in this last example that, since the coefficients are
pure numbers, it does not matter whether we work in units of three years
or of one year.

Measures of skewness

7.9 The departure of a frequenc 5 ^-distribution from symmetry has a
certain interest, and several measures have been devised to permit of the
measurement of this skewness. Such measures should {a) be pure numbers,
so as to be independent of the units in which the variable is measured,
and (6) be zero when the distribution is symmetrical.

7.10 Three such measures des^ve mention. In the first place, we can
define

V 2Q

- (7.14)

MOMENTS, SKEWNESS AND KURTOSIS

i6r

This can be put in the form —

Skewness =

{Q,-Mi)+{Mi~Q,)

. (7.15)

i.e. the skewness is taken to be the difference of the quartile deviations from
the median divided by their sum. It is clearly a pure number, for both
numerator and denominator have the same dimensions, and it is zero when
the distribution is symmetrical. It varies from —1 to +1.^

This is a rather rough-and-ready measure which might, however, be
useful if we were using the semi-interquartile range as a measure of dis-
persion and were unable or unwilling to calculate the standard deviation.

7.11 The most common measure of skewness is Pearson's, defined by

Skewness

Mean*— Mode
Standard deviation

. (7.16)

This evidently is a pure number and is zero for symmetrical distribu-
tions.

7.12 The calculation of this coefficient of skewness is subject to the
inconvenience of determining the position of the mode. We may circum-
vent this difficulty in several ways. In the first place, for distributions
which are obviously not too skew we may use the empirical relation
of 5.27. We then have —

. . . (7.17)

Standard deviation

Secondly, for a large class of curves to wliich the moderately skew
humped curve is a close approximation, the skewness of equation (7.16)
is given exactly by

Skewness = .... (7.18)

We may, therefore, take this to be an approximation to the value given by
equation (7.16).

It should be noted that the measures (7.14) and (7.16) are positive if
the longer tail of the distribution lies toward the higher values of the
variate (the right) and negative in the contrary ca*^. This accords with
the anticipator}^ remarks of 4.20. The measure (7.18) is to be regarded
as without sign.

^ In the loth and previous editions of this book the measure Skewness

was siiggested, i e. twice the measure {7 14). The above form has the advantage that
Its limits are i and 1 .

i 62

THEORY OF STATISTICS

LImiB of the measmes of skewness

7.i3 We have already remarked that the measure given by equation
(7.14) lies between —I and +1. There is no limit in theory to the measure
(7.16) or its approximation (7.18), and this is a slight drawback. But
in practice the value given by equation (7.16) is rarely very high, and for
moderately skew single-humped curves is usually less than unity.

It has been shown that the quantity

Mean— Median
Standard deviation

lies between

the limits —1 and +1, and the measure (7.17) therefore lies between —3
and +3. In practice it rarely approaches these limits.

Example 7.7. — Let us once again consider the height distribution of
Table 4.7, which has been already discussed in this chapter (Examples 7.1,
7.3 and 7.5).

We have—

Mean (Example 5.1, p. 106) =67*46 inches

S.d. (corrected, Example 6.4, p. 134) = 2*56 inches

Median (Example 5.3, p. 112) =67*47 inches

(Example 6.9, p. 141) =65*71 inches

^3 (ibid,) =69*21 inches

Q (ibid,) = 1*75 inches

(corrected. Example 7.5, p. 160) = 0*000155
(ibid,) = 3*149

The measure of skewness (7,14) is, then,

Sk == ^^i

_ 65*71 +69*21 -(2 x 67*47)

2x1-75

= -0*006

We can clearly place no reliance on this figure. The median and
quartiles were obtained by methods of approximation which we cannot
expect to give accuracy to the second decimal place. We can only
conclude, therefore, that so far as the measure (7.14) is concerned, there
is no significant skewness.

The measure (7.18) gives —

cp 0*0124x6*149
”2(15*745-0*001-9)

_ 0*0124x6*149
2x6*744
= 0*006

Here again the skewness is extremely small, and is, in fact, almost
equal to the value given by (7.14).

MOMENTS, SKEWNESS AND KUETOSIS

163

If we take the measure (7.17) we get —
SJ, - 3(M-Mt)

^ -0 03
2-56

= - 0-012

This value is suspect because we have determined the mean and the
median only to the second decimal place, but clearly the value is small.

We conclude that there is only very slight skewness. At this stage we
cannot say whether such small skev^mess is significant, but it is at least
probably attributable to sampling fluctuations.

Example 7.8. — For the marriage data of Examples 7.2, 7.4 and 7.6
it vill be found that, using the working mean as origin —

Mean =: 0-2944

Median =* —0-4018
= -1-4568
1*2316

and

cr (corrected) (Ex. 6.5) =2-6408
^ 1 = 3-854
^2 =8-333

The measure (7.14) is —

_ 1-6334-1 -0550
~ 1-6334-1-1 -0550
_ 0-5784
2-6884

= 0-22

The measure (7.18) is —

_ v'3-854(ll-333)

2(41-665 -23-124 -9)

_ 1-963x11-333
2x9-541
= 1-17

The two are very different, as we might expect, but both indicate
strong positive skewness. As a matter of interest we may compare the
value (7.17), which gives

3x0-6962

2-6408

164

THEORY OF STATISTICS

Kurtosis

7.14 The coefficient ox its derivative 72 is used to measure a property
of the single-humped distribution known as kurtosis {Kvprds, humped).

We take as the standard value of number 3, for reasons which

will appear when we study the so-called normal ” curve (8.24). This
curve is approximately of the shape given in fig. 4.5, page 81. Curves
with values of l^ss than 3 are called platykurtic {nXccrijs, broad, -f
Kvprds). Curves with values greater than 3 are called leptokurtic^
(AeTTTds, narrow, +Kvp76s), ** Student '' gives an amusing mnemonic for
these names : Platykurtic curves, like the platypus, are squat with short
tails. Leptokurtic curves are high with long tails like the kangaroo —
noted for “ lepping '' 1

Example 7.9. — In the height distribution of Examples 7.1, 7.3, 7.5
and 7.7— ^2 = 3-149

r« =^2-3 =0-149

Hence the curve is slightly leptokurtic.

On the other hand, in the marriage distribution of Examples 7.2, 7.4,

7.6 and 7.8— = 8-333

7, = 5-333

and the curve is very leptokurtic.

Cumulants

7.15 We may conclude this chapter by referring briefly to a set of
quantities similar to moments which have some theoretical and practical
importance. These are the cumulants.^

The cumulants are defined by a rather complicated mathematical
expression which we shall not here reproduce. For present purposes it
is sufficient to note that the first four cumulants may be expressed as
simple functions of the first four moments. In fact we have —

ATg

Xz = — 3/<l >2 ' +2/ti

*-•4 = i“4 ' -4/<i >3 ' -S/t,'® 4- 1 ' >2 ' -6/^1

^ These terms are due to Karl Pearson and appear to have been given for the first
time in Biomeinhat 1905, 4, 169. By a slip Uptokurtcsis is there inadvertently applied
to distributions for which y?3<3.

It has often been stated that platykurtic curves are relatively more flat- topped and
leptokurtic curves more peaked than the *' normal curve. This is the origin of the
name and of Student's " mnemonic, and the assertion was made in the 13th and earlier
editions of this book. It is, however, very difficult to justify in general.

* These quantities were introduced into statistics by T. N. Thiele under the name of
semi-invanants, the forms ** seminvariant ” and “ half-invariant " also occurring in
earlier literature. The word cumulant “ is preferable and is now m general use,
there being other families of quantities which also have the seminvariant propeity m
the algebraical sense.

MOMENTS, SKEWNESS AND KURTOSIS

165

In particular, about the mean,

=0 \

I . . . . (7 20)

ATg — fl^ I

AC 4 =/<4-3/tij2 J

7.16 These relations are used in the calculation of the cumulants, the
moments being first ascertained in the manner of the earlier sections of
this chapter. For instance, the first four cumulants of the height dis-
tribution which has served us as an example are, about the mean,

= 0

^2 =6*616805

=- -0-207839

= 137-689185-3X (6*616805)2 =6-34286
if we take uncorrected values of the moments.

7.17 The cumulants have several remarkable properties. In the first
place, all cumulants except the first are independent of the origin of
calculation. The moments vary according to the point about which
they are calculated, which makes it necessary to specify the origin A
in speaking of them. The cumulants, on the other hand, do not, so that
it is unnecessary to specify any value A in giving their values ; the sole
exception to this rule is the first cumulant, which is the same as the
first moment

Secondly, if the scale of measurement of the variate is altered by
multiplying all values by a constant a, the nth cumulant is multiplied
by Thus, in the height distribution, if we change our scale to centi-
metres instead of inches, and so multiply all values of the variate by 2-54,
the cumulants in the previous section are to be multiplied by 2*54, 2*54^,
2*54^ 2-54^ respectively.

We shall also see in the next chapter that the cumulants take simple
values for certain theoretical frequency-distributions of importance.

SUMMARY

1. The wth moment about the point A is defined as

where ^=X—A, and X is the value of the variate.

2. The nth moment about the mean is written

i66

THEORY OF STATISTICS

3 /u =~ V 2 — • • • +(““1)'*^^"

where

d =: M -A

and in particular

J^s =/^3'-3i/^2'4'2i^

/I 4 = /i4^ --4djUQ' ■^ 6 (i^/i 2 ' — 3d^

4, Sheppard’s corrections for the moments are —
//.g (corrected) —

X ^

(corrected) =/^3

tii (corrected) =

y% — /^2~"3 —

6 Pearson’s measure of skewness is given by

Sk =

Mean— Mode
Standard deviation

which, for a large class of curves, is equal to

yA(^2+3)

2(5^2 -6^1 -9)

7. If the standard deviation is not known, a rough measure of skewness
is obtained by taking

Sk =

8. Distributions for which >^2> 3 are said to be leptokurtic ; those for
which y?2< 3 are platykurtic.

9. The first four cumulants, in terms of the moments about the mean,
are —

== 0

/Cg =/i2

/C 3 =/43

^^4 =/‘4-3/«a*

10. The cumulants are independent of the origin of calculation, except'
the first, which is equal to the mean.

MOMENTS, SKEWNESS AND KURTOSIS l6j

EXERCISES

7.1 Find the first four moments about the mean of the distribution of
males in the United Kingdom according to weight given in Exercise 4.6.,
page 100. (Correct your values for grouping.)

Hence find and and measure the kurtosis of the distribution.

7.2 For the same distribution find the three measures of skewness,
approximating to the mode by the empirical relation of 5.27.

7.3 Find the first four moments about the mean, the values of /? 2 »
and the three measures of skewness for the following distribution (see
table below). (Apply Sheppard's corrections.)

7.4 In the data of Example 7.1, group the individuals by intervals of
three inches (57-, 60-, etc.) and calculate the first four moments about
the mean. Compare your results with those of Example 7.1. (a) before
Sheppard's corrections are applied, and (b) after Sheppard’s corrections
are applied.

7.5 Find the third and fourth moments about the mean of the binomial
series —

qn^ nq»-^p,

n[n—\)

1.2

. . where

(continuing the work of Exercise 6.10, page 149).

Data for Exercise 7.3 — 4912 Cows classified according to tfeeir yield of milk

(Data from J. F. Tocher, ** An Investigation of the Milk Yield of Dairy Cows,*'
Btometnka, 1928, 20B, 10.5.)

Yield of milk
(gallons per week)
(Central value of
interval)

Number of
cows

Yield of milk
(gallons per week)
(Central value of
interv^al)

Number of
cows

2.3

214

153

112

151

236

339

30 1

499

552

585

586

496

448

Total

4,912

284

i68

THEORY OF STATISTICS

7.6 The first four moments of a distribution about the value 4 are — 1 ' 5,
17, —30 and 108 ; find the moments about the mean and the origin.

7.7 Show that for a symmetrical distribution all moments about the mean
of odd order are zero.

7.8 Show that for any distribution

7.9 Calculate the second, third and fourth cumulants of the distribution
of Australian marriages of Example 7.2, [a) from the moments about the
mean, using equation (7.20), and {h) from the moments about the value
28*5, using equation (7.19) ; and hence verify that the values of the
cumulants are independent of the origin of calcidation. (Use uncorrected
values of the moments.)

7.10 Show that

a =\/^2

CHAPTER EIGHT

THREE IMPORTANT THEORETICAL
DISTRIBUTIONS

THE BINOMIAL, THE NORMAL AND THE POISSON

Theoretical distributions

8.1 In the examples of frequency-distributions which we have given
in Chapter 4 and subsequent chapters we have been careful to take data
from observation and experiment. It is possible, however, starting with
certain general hypotheses, to deduce mathematically what the frequency-
distributions of certain populations should be. Such distributions we
shall call theoretical.

8.2 There are three theoretical distributions which, from their historical
interest as Well as their intrinsic importance, occupy a position in the
forefront of statistical theory. They are, in the order of their discovery,
the Binomial (due to James Bernoulli, circa 1700), the Normal (due to
Demoivre, but more often associated with the names of Laplace and
Gauss, who discussed it at the close of the eighteenth and the beginning
of the nineteenth centuries), and the Poisson (due to S. D. Poisson, who
published it in 1837).

These three are, so to speak, the classical distributions. Certain others
were discovered during the nineteenth century, but it was not until the
end of the century that there began the second period of statistical dis-
covery which has since given us a wealth of theoretical distributions. Even
this latest crop depends to some extent on the properties of the first three,
and particularly of the Normal Distribution The three therefore form,
historically and logically, the starting-point of the theory of particular
distributions, and in this chapter we propose to give an account of their
main properties.

The binomial distribution

8.3 If we may regard an ideal coin as a uniform, homogeneous circular
disc, there is nothing which can make it tend to faU more often on the
one side than on the other ; we may expect, therefore, that in any long
series of throws the coin will fall with either face uppermost an approxi-
mately equal number of times, or with, say, heads uppermost approximately
half the times. Similarly, if we may regard the ideal die as a perfect
homogeneous cube, it will tend, in any long series of throws, to fall
with each of its six faces uppermost an approximately equal number of

x 69

170

THEORY OF STATISTICS

times, or with any given face uppermost one-sixth of the whole number
of times. These results are sometimes expressed by sajdng that the chance
of thromng heads (or tails) with a coin is 1 /2, and the chance of throwing
six (or any other face) with a die is 1/6. To avoid speaking of such
particular instances as coins or dice we shall in future, using terms which
have become conventional, refer to an event the chance of success of
which is p and the chance of failure q. Obviously

8.4 We will now assume that the events in a number of tiials are all
independent, i.e. that the chances p and q are the same for each event
and remain constant throughout the trials. The case corresponds to the
tossing of perfect coins or the throwing of perfect dice.

Suppose now we take a number of sets of n trials and count the number
of successes in each set ; for example, we might toss a coin ten times for
each set, and observe the number of heads in each set of ten. In general,
there will be some sets with no successes, some with one success, some with
two successes, and so on. Hence, if we classify the sets according to the
number of successes which they contain we shall get a frequency-dis-
tribution. Table 4.15, page 96, gives such a distribution for some dice-
throwing experiments. We shall now see how, on the assumption of
independence of successive events to which we have just referred, the
nature of this distribution may be theoretically determined.

8.5 For the case of single events we expect in N trials to get Np successes
and Nq failures.

Suppose now we take N pairs of events, i.e. two to the set. There will
be Nq cases in which the first event is a failure, and, in virtue of the in-
dependence of the events, among these Nq there will heNqxq failures, and
Nqxp successes, of the second event on the average. Similarly, of the Np
cases in which the first event was a success, the second event will, on the
average, be a success in Np xp and a failure in Np x q cases. Hence there
will be Nq^ cases in which both events are failures, 2Npq cases with one
success and one failure, and Np^ cases in which both are successes.

If we now take N sets of three events, we see that, of the Nq^ cases in
which the first two events were failures, Nq^xq will give a third failure
and Nq^xp one success ; of the 2Npq cases, 2Npq^ will give two failures
and a success and 2Np^q one failure and two successes ; and of the Np^
cases, Np^q will give one failure and two successes and Np^ will give three
successes. Hence the number of sets with 3 failures, 2 failures and 1
success, 1 failure and 2 successes, and 3 successes are, respectively,

Nq\ SNq^p, SNqp^ Np^

8.6 From these results it is evident that the frequencies of 0, 1, 2, . . .
successes are given

for one event by the binomial expansion of N{q+p)
for two ^yents „ „ „

for fAm events „ „ „

THREE THEORETICAL DISTRIBUTIONS

I7I

In general, for n events the frequencies of successes in N sets are given
by the successive terms in the binomial expansion of N{q+pY^ i.e.

Mn -1)^_ 2^2 , w(w-l)(w - 2)

1.2 " " ■ 1.2.3

This is the so-called binomial distribution.

Example 8.1. — If we take 100 sets of 10 tosses of a perfect coin, in
how many cases should we expect to get 7 heads and 3 tails ?

Here p=i,

Hence, the numbers of succes.se5 0, 1, . . . 10 are the terms in 100(-H-i)^<*,

i.e.

100

The term giving 7 successes and 3 failures is —

lOOx^oC.a)’^)®

10.9 8 1

= 100 .

1.2.3 ■2“

3000

256

=12 approximately.

Example 8.2. — In the previous example, in how many cases should
we expect to get 7 heads at least ? As before, the numbers of successes
are the terms in

100

210

1 + 10 -+

10.9

1.2

We require the sum of terms with 7, 8, 9, 10 successes,
number is, then.

Our expected

100[10.9.8 10.9 I 10 1

2io| 1.2.3'^ 1.2"^ 1

100

210

{176}

_1100

5=17 approximately.

172

THEORY OF STATISTICS

Genera] form of the binomial distribution

8,7 The form of the binomial distribution depends (1) on the values
of p and q, (2) on the value of the exponent n.

If p and q are equal the distribution is evidently symmetrical, for p
and q may be interchanged without altering the value of any term, and
consequently terms equidistant from the two ends of the series are equal.

If, on the other hand, p and q are unequal, the distribution is skew.
The following table shows the calculated distributions for ft =20 and
values of p, proceeding by 0*1, from 0*1 to 0*5. When ^=0*1, cases of
two successes are the most frequent, but cases of one success almost
equally frequent : even nine successes may, however, occur about once
in 10,000 trials. As p is increased, the position of the maximum frequency
gradually advances, and the two tails of the distribution become more
nearly equal, until />=0*5, when the distribution is symmetrical. Of
course, if the table were continued, the distribution for =0*6 would be
similar to that for j'=0*6, but reversed end for end, and so on.

TABLE 8.1 — Terms of the binomial series 10,000 for values ofp from 0*1 to 0*5

(Figures given to the nearest unit)

Number of

*=0-2

#>=0-3

^=0-4

p=0*5

successes

5=0*9

5=0*8

^=0*7

5=ss0 *6

^=0*5

1,216

115

2,702

576

—

2,852

1,369

278

1,901

2,054

716

123

898

2,182

1,304

350

319

1,746

1,789

746

148

1,091

1,916

1,244

370

545

1,643

1,659

739

222

1,144

1,797

1,201

654

1,597

1,602

—

308

1,171

1,762

—

120

710

1,602

—

355

1,201

—

146

739

—

— i

370

—

— i

—

148

—

— .

—

8.8 If p^q, the effect of increasing n is to raise the mean and increase
the dispersion. If p is not equal to however, not only does an increase
in n raise the mean and increase the dispersion, but it also lessens the
asymmetry ; the greater for the same values of p and g, the less the

THREE THEORETICAL DISTRIBUTIONS

173

as 5 rmmetry. Thus, if we compare the first distribution of the above table
with that given by «=100, we have the following —

TABLE 8.2— Terms of the binomial series 10,000 (0-9+0 1)''>'>
(Figures given to the nearest unit)

Number

successes

Frequency

Number

successes

Frequency

Number

successes

Frequency

1,148

193

1,304

106

1,319

3 i

1,199

159

988

339

13 1

743

596

513

S89

327

ng. 8 . 1 .— Frequency-polygons the binomial (0*94-0-l)* for various values of w

174

THEORY OF STATISTICS

The maximum frequencies now occur for 9 and 10 successes, and the two
tails '' are much more nearly equal. If, on the other hand, n is reduced
to 2, the distribution is —

Number of
successes

Frequency

8,100

1,800

100

and the maximum frequency is at one end of the range.

The tendency towards symmetry may be seen from fig. 8.1, in which
the binomial (0*9 +0-1)" has been drawn for various values of n. See
also 8.12 below.

Constants of the binomial distribution

8.9 We proceed to find the lower moments of the distribution N{q+p)**,
Taking an arbitrary origin at 0 successes, we have the successive
deviations g as 0, 1, 2, . , . n, and hence,

. . .+{pnxn)

=^np{q+pY-'^

Now, qJ^p^i

Hence, jii^^=^np

That is, the mean M is np.

We have, further,

/^2'==(9”X0)+("Cig«~ij5>xl)+K2f»~2^2x22)+ . . . +{p»xn^)

— -q^-^p^+ . . .

The expression in brackets is the first moment of the binomial (q+p)*^^
about origin —1, and hence is equal to (^~1) j^>4-l.

Hence,

fi.^'=::np{{n~l)p+l}

It may also be shown in a similar way (but we omit the proof) that
fii =nf{{n—l){n—2)p^+3{n—l)p+iy

jiit=‘np’^{n—l)(fir—2){n—S)p^+&{n—l){n—2)p^+7{n—l)p-^iy

THREE THEORETICAL DISTRIBUTIONS 175

8.10 From these results we may find the moments about the mean
We have —

=Kp{{n—\)p+l').-n^p^

=np{l~p)

=^npq

Hence we have the important result that —

( 8 . 1 )

8,11 Similarly, it will be foimd that —

Hence,

liz=npq{q-p) .

—Qpq) .

I l-6j>g

Pqn

( 8 . 2 )

(8.3)

(8.4)

(8.5)

8,12 Thus the binomial distribution has mean np and standard deviation
"s/npq. It is instructive to note that and are both of order 1

Hence, as n becomes larger, the distribution tends to symmetry and
zero kurtosis.

The values of and some values of p and q and ranges of n are

shown in Tables 8.3, 8.4 and 8.5.

From an inspection of these tables it will be seen that even for an
extremely small value of p the binomial tends to zero and zero kurtosis
for values of n well within practical limits. For the symmetrical binomial
^=^=0*5, is of couise zero, and §2 rapidly approaches 3.

TABLE 8.3.— -Values of and for the binomial with ^==0 02, ^-0*98

(From M. Greenwood, Bumdrika^ 1313, 9, 69.)

100

0-4702

3-4502

200

0-2351

3-2251

300

0*1567

.3-1501

400

0-1176

3-1126

500

0-0940

3-0900

600

0-0784

3-0750

700

0-0672

3 0643

800

0-0588

3-0563

900

0-0522

3-0500

1,000

0-0470

3-0450

176

THEORY OF STATISTICS

TABLE 8.4* — Values of and the binomial with 5'=(l-9

fix

100

0-0711

3-0511

200

0*0356

3-0256

1,000

0-0771

3-0051

Mechanical representation of the binomial distribution
8.13 There is an interesting mechanical method of constructing a repre-
sentation of the binomial series. The apparatus, which is illustrated
in fig. 8.2, consists of a funnel opening into a space — say a J inch in depth
— between a sheet of glass and a back-board. This space is broken up by

successive rows of wedges like 1, 2 3,

Fig. 8.2. — The Pearson-Galtoa
binomial apparatus

4 5 6, etc., which will divide up into
streams any granular material such as
shot or mustard seed which is poured
through the funnel when the apparatus
is held at a slope. At the foot these
wedges are replaced by vertical strips,
in the spaces between which the
material can goUect. Consider the
stream of material that comes from
the funnel and meets the wedge 1.
This wedge is set so as to throw q parts
of the stream to the left and p parts
to the right (of the observer). The
wedges 2 and 3 are set so as to divide
the resultant streams in the same
proportions. Thus wedge 2 throws
parts of the original material to the
left and qp to the right, wedge 3 throws
pq parts of the original material to
the left and p^ to the right. The
streams passing these wedges are
therefore in the ratio of q ^ : 2qp : p^.
The next row of wedges is again set
so as to divide these streams in the

THREE THEORETICAL DISTRIBUTIOKS

177

same proportions as before and the four streams that result will bear the
proportions (f : 3q^p : 3qp^ : p^. The final set, at the heads of the vertical
strips, will give the streams proportions q ^ : 4q^p : 6q^p ^ : 4qp ^ : p^, and these
streams will accumulate between the strips and give a representation of the
binomial by a kind of histogram, as shown. Of course as many rows of
wedges may be provided as may be desired.

This kind of apparatus was originally devised by Gallon in a form
that gave roughly the symmetrical binomial, a stream of shot being
allowed to fall through rows of nails, and the resultant streams being
collected in partitioned spaces. The apparatus was generalised by Karl
Pearson, who used rows of wedges fixed to movable slides, so that they
could be adjusted to give any ratio oi q : p,

8.14 It must not be forgotten that although we have spoken in 8,12 of
the skewness and kurtosis of the binomial distribution, it is essentially
discontinuous. This is a serious limitation.

Consider, for example, the frequency-distribution of the number of male
births in batches of 10,000 births, the mean number being, say, 5,100. The
distribution wiU be given by the terms of the series (0*49+ and
the standard deviation is, in round numbers, 50 births. The distribution
will therefore extend to some 150 births or more on either side of the mean
number, and in order to obtain it we should have to calculate some 300
terms of a binomial series with an exponent of 10,000 I This would not
only be practically impossible without the use of certain methods of
approximation, but it would give the distribution in quite unnecessary
detail : as a matter of practice, we should not have compiled a frequency-
distribution by single male births, but should certainly have grouped our
observations, taking probably 10 births as the class-interval. We want,
therefore, to replace the binomial polygon by some continuous curve,
having approximately the same ordinates, the curve being such that the
area between any two ordinates and will give the frequency of
observations between the corresponding v^ues of the variable % and

Limiting form of the binomial for large n

8.15 When n becomes large, each term of the binomial becomes small.
We are, however, concerned with the sum of the terms falling wdthin
certain ranges, and these will not be small in general.

Let us consider first of all the case when p and q are equal. The terms
of the series are —

The frequency of m successes is

n !

178

THEORY OF STATISTICS

and the frequency of m+1 successes is derived from this by multiplying
it by (^— m) /(m+l)* The latter frequency is therefore greater than the
former so long as

n—1

Suppose, for simplicity, that n is even, say equal to 2k ; then the frequency
of k successes is the greatest, and its value is

( 8 . 6 )

The polygon tails off symmetrically on either side of this greatest ordinate.
Consider the frequency of k +x successes ; the value is

and therefore

(2^)!

{k-\-x) 1 (k—x) *

(8.7)

y,_ {k)(k-l){k~2) . . . (A-;t+l)
yo (^ + 1 )(^+ 2 )(^+ 3 ) . . . {k-^x)

(■riX'-iX'-S)

( 8 . 8 )

Now let us approximate by assuming that k is very large, and indeed
large compared with x, so that {x jk)^ may be neglected compared with
(xjk). This assumption does not involve any difficulty, for we need not
consider values of x much greater than three times the standard deviation
Or SVk 12, and the ratio of this to k is 3 jV^ which is necessarily small
if k be large. On this assumption we may apply the logarithmic series

^3 ^4

iog.(i+<y)=<j~^+|-"-+ . .

to every bracket in the fraction (8.8), and neglect all terms beyond the
first. To this degree of approximation,

lOgtf — = -.-{1 +2+3+ • • * -\-X — l)—:g
JVo ^ «

THREE THEORETICAL DISTRIBUTIONS

179

Therefore, finally

-L* - £l

yx=y^e * . (8.9)

where, in the last expression, the constant k has been replaced by the
standard deviation a, for 0 ^=^ /2.

8,16 The case when p is not equal to q may be treated in a somewhat
similar way but is slightly more complicated.

As before the frequency of m successes is

n I

m ! {n—m)

•qp~^ ^pi^

The frequency of (m+1) successes is derived by multiplying this

expression by

n—m p

^ and hence is greater than the former if
m+1 q

n— w p ..
m<np^q

Let us assume that np is a whole number. Since n is going to tend
to infinity, this really imposes no limitation on our work.

The maximum frequency is, then.

yo^N-

(np) I {nq)

-q^p^P

The frequency of pn successes is

. ( 8 . 10 )

Hence,

rtf '•

{np-{-x)\{nq—x)\

ys_ np\nq\
ytTi^+x ) ! {nq—x ) !

qnq-^xpnp-^-x

. ( 8 . 11 )

. ( 8 . 12 )

Now, by an important theorem due to James Stirling (1730), if « be large,
we have approximately

n ! =A/2»7r»»«-*

i8o

THEORY OF STATISTICS

Applying this formula here —

V2np7i{np)^Pe- ^pV 2nqn{nq)^e-^p^
yo V2{np -{-x)n{np 2{nq-—x)7r[nq--x)**^-*e-^~^^q^

which reduces to

yx_ 1

Hence,

fi+— \

= -\np+x+\

V— - —

}\np 2w®

2n^p^ 3n^p^

After a little rearrangement this becomes —
log* I

■■■)

2n^q^ 3n^q^ , )

\ X^{p^+q^)

-t±x.

) 2npq An^p^q^

2npq

%n^p^q^

+ terms of order -4 and higher

Since we have, neglecting the terms of order and higher,

which are small compared with the others when n is large-

log^

xHP^+q\s—P ( ~ ^

2npq An^p^q^ '^np(^

3npqj

(8.13)

Put, as before, npq^Q^y where a is the standard deviation of the
binomial. If n be large, the second term is small compared with the first.

Further, since we need not consider values of — much greater than 3,

be small, we can neglect the whole of the third term. On these

Vnpq

assumptions we have —

log

yo^

yx=-yo^

- —
2a®

(8.14)

as before.

THREE THEORETICAL DISTRIBUTIONS

i8i

The expression is merely V and so we have in effect simply

ass^ed small ; however much p and q differ we can always make
as small as we please by increasing n sufficiently.

8.17 Hence, whether or not p is equal to q, the binomial distnbution
tends to the form* of the continuous curve ((8.9) and (8.14)) when n
becomes large, at least for the material part of the range. As a matter
of fact, the correspondence between the binomial and the curve is sur-
prisingly close even for comparatively low values of provided that
p and q are fairly near equality. The student may care to ciraw the curve
with the aid of the tables given at the end of this book (see below, 8.26)
and compare it with some of the simpler binomials drawn to the same
scale.

8.18 The curve

y=yo&

is called the normal curve. A population classified according to a con-
tinuous variate whose ideal frequency-distribution is a normal curve is
called a normal population.

The applications of the normal curve are by no means limited to dis-
tributions of the binomial type. Before we refer to its many practical
and theoretical applications, however, we shall give a short account of
its main properties.

Properties of the normal curve

8.19 The normal curve is obviously symmetrical about the point ^^=0,
for its equation is independent of the sign of x. At this point the
ordinate has its maximum value. The mean, the median and the mode
coincide, and the curve is, in fact, that drawn in hg. 4.5, page 81 , and taken
as the ideal form of the symmetrical curve.

8.20 The curve is specified completely by defining the mean (the origin
of x), the standard deviation cr and the value ^0.

In actual practice, as, for example, when we are trying to fit a normal
curve to given data, we are not given yQ itself, but have to calculate it
from the fact that the area of the curve must be equal, on the chosen
scale, to the total number of observations. For this reason we wish to
find the area under the curve

i 82

THEORY OF STATISTICS

8.21 From 4.14 it will be seen that the area of a histogram, that is to
say, the total number of observations which it represents, is given by

r~n

Area— E {Jr) X h

where h is the width of the interval, fr is the frequency in the rth interval
and there are n intervals

As the histogram tends towards the continuous curve the width of the
intervals becomes smaller and the number of terms in the summation
becomes larger. For the normal curve, which extends to infinity on
either side of the mean, the limit to which the sum tends as the intervals
become indefinitely small and the number of terms indefinitely large is
written

\_^y,e^'dx

the sign / being a conventional form of the summation sign S and dx
representing the infinitesimally small value of h.

This is the notation of the integral calculus, and the quantity | F{x)dx

IS said to be the integral of F{x) with respect to x between the limits —a
and In this book we shall not use the methods of the integral calculus,
and accordingly it will be necessary for us to state certain results without
proof. It will be sufficient if the student bears in mind that the process of
integration is one of proceeding to the limit in cases of straightforward
summation with which he is already familiar.

8.22 The area of the

curve

;g»

is then

TOO

]_^yoe^^dx

and this is equal to

y^P X =2 • 506627y<

Hence the curve

a^/2n

has unit area, and for this reason the equation of the normal curve is usually
written in the standard form

1 -
y=:

. (8.15)

THREE THEORETICAL DISTRIBUTIONS

183

From this the form corresponding to a distribution of any given frequency
is immediately written down. In fact, if the frequency is N, the corre-
sponding normal curve is

N -
a\/27r

-fL

2o*

( 8 . 16 )

Constants of the normal curve

8,23 The mean of the curve is, as we have seen, located at the origin.
If we wish to WTite the curve with reference to some other point as origin,
we can do so in the form

y=-

— =e

. ( 8 . 17 )

where m is the excess of the mean over the value chosen as origin.

The standard deviation of the curve is 0 *, and the variance is accordingly
a^.

The higher moments are calculated by the processes of the integral
calculus. Since the nih moment about the mean is given by

we have, proceeding to the limit, that the ^th moment of the normal curve
is

1 - *1
crv^27rJ-.co

If n is odd this vanishes, as it must for any S3rmmetrical curve,
we have —

J^ln

If n is even
. (8.18)

and hence,

4 . 3.2

2 . 2.2

=3a^

( 8 . 19 )

8.24 From these results it follows that —

A 2 — 72 ““Oj

. ( 8 . 20 )

i.e. the normal curve has zero kurtosis. This is, in fact, the origin of the
choice of the apparently arbitrary value 3 in the definitions of platy- and
lepto-kurtosis (7.14).

We may also state without proof the important result that all cumulants
of the normal curve of orders higher than the second vanish identically.

184 THEORY OF STATISTICS

8.25 The mean deviation of the norma] curve is —
a^|=0-79788 . . . cr

This is the origin of the rule given in 6.22, that the mean deviation is
approximately | of the standard deviation. The result is true of the
normal curve, and very approximately true of curves which do not differ
markedly from the normal form. The rules that a range of 6 times the
standard deviation includes the great majority of the observations (6.13)
and that the quartilfe deviation is about f of the standard deviation (6.25)
were also suggested by the properties of the normal curve (see below,
8.28 and 8.29).

Ordinates of the normal curve

8.26 The normal curve is so important that tables have been prepared
to give (1) the ordinate of the curve corresponding to any given value

of X, i.e. the values ^ (^) areas of the curve to the

1 f«

right and the left of any given ordinate, i.e. the values of ■'"7— e Hx

’V^TTjx

1 -i?

and“~7= e Hx. Table 1 of the Appendix gives the values of the

2?t J ^ qq

ordinate for values of x proceeding by steps of one-tenth of the standard
deviation. The values are, of course, the same for positive as for negative
values of x. More extended tables will be found in Tables for Statisticians
and Biometricians, Part L

The ordinate of any normal curve corresponding to a specified value of
the variate is easily obtained from the table, as may be seen from the
following example —

Example 8.3. — ^To find the ordinate of the normal curve given by —

10,000

W27T

corresponding to the variate value x^l.

Here

i\r=10,000, a=4

Altering the value of cr is equivalent to altering the scale of x. The
ordinate in this curve corresponding to a? = 7 will be the same as the ordinate
of the curve of unit s.d. corresponding to x^\ =1*75,

From Appendix Table 1, when

x=^l*8 y =0*07895
^^1.7 j;==0*09405

THREE THEORETICAL DISTRIBUTIONS

185

Hence, by simple interpolation, when

^= 1.75 y =0-08650

The ordinate is 10,000 /4 times this, i.e. is equal to 216. This is accurate
to the nearest unit.

Area of the normal curve — the probability integral
8.27 A table of the areas of the normal curve cut off by ordinates at
specified values of x is given in Table 2 of the Appendix. As in the case
of the table of ordinates, this table is applicable to all normal curves,
whatever the value of their standard deviation, the areas cut off on

\^2it

e ^ by ordinates at a; being the same as those cut off on y =

1 -

2cr*

aV27r

by ordinates at More extended tables will again be found in Tables for
a

Statisticians and Biometricians, Part /.

The area of the normal curve to the left of the ordinate at x or, it may
be, between the ordinates at 0 and x — conventions differ — is sometimes
termed the probability integral or the error function. These names arise
from the use of the function in the theory of sampling and the theory
of errors respectively.

Example 8.4. — Find the frequency represented by the smaller area of
10 000

the curve 'y = — '-=e ^2 cut off by the ordinate at ^==7.

4V27r

Here

(x=4, ^=1-75

For -==1-75=1*5 +0 • 25 the table gives the value 0 • 9599. Hence the
a

smaller fraction equals 1 --0* 9599 =0-0401 and multiplying this by 10,000,
we have the frequency represented, i.e. 401.

Example 8.5. — A hundred coins are thrown a number of times. Ho?/
often approximately in 10,000 throws may (1) exactly 65 heads, (2) 65
heads or more, be expected ?

The number of heads is given by the terms in

10,000(Ki)'°®

The standard deviation is — =2,000, and the

^ (j

exponent 'is large enough for us to be able to take the distribution as
normaL

i86

THEORY OF STATISTICS

The mean number of heads is 50, and 65— SO—Scr. The frequency of a
deviation of 3a is given at once by Appendix Table 1 as 2,000x0*00443
=8*86, or nearly 9 throws in 10,000 A throw of 65 heads will therefore
be expected about 9 times.

The frequency of throws of 65 heads or more is given by Appendix
Table 2, but a little caution must now be used, owing to the discontinuity
of the distribution. A throw of 65 heads is equivalent to a range of
64 * 5-65 • 5 on the continuous scale of the normal curve, the division between
64 and 65 coming at 64*5. 64*5 — 50 = +2* 9a, and a deviation of
+2* 9a or more will only occur, as given by the table, 187 times in 100,000
throws, or, say, 19 times in 10,000.

8.28 From the table of areas we can find approximately the position

of the quartiles. In fact, we require the value of - which will give us 0 • 75

as the greater fraction of the area. From the table we see that this value
must lie between 0*67 and 0*68. Simple interpolation gives

|o-67+0-01^j =0-675

a more exact result is

Quartile deviation =0* 67448975a . . . (8.21)

This is the origin of the rough rule that the semi-interquartile range is
usually about | of the standard deviation.

8.29 We also observe from the table that an ordinate 3a from the mean
cuts off an area 0 • 99865 of the whole. The smaller fraction left is therefore
0*00135 of the whole. Since the curve is symmetrical, it follows that
a range of 3a on each side of the mean will cut off all but twice this, i.e.
all but 0*00270 of the whole. This again is the origin of the rule that
such a range includes the great majority of the observations.

The normal distribution as an error distribution

8.30 We have deduced the normal distribution as a limiting form of
the binomial distribution when n, the exponent, is large. This however,
is only one of the ways in which the normal curve occurs in statistical
literature, and Gauss was led to it by a totally different line of reasoning,
via. by inquiring what law of distribution errors of observation should
obey in order to make the arithmetic mean of a set of measurements the
most likely value of the '' true magnitude.

THREE THEORETICAL DISTRIBUTIONS

187

8.31 Suppose we take a population of measurements of some magnitude,
and consider the population of deviations from the true value. Let us
further suppose that any deviation is the result of the operation of an
indefinitely large number of small causes, each producing a small perturba-
tion. Let us assume that the small perturbations are all equal, and that
positive and negative perturbations are equally likely.

Then it may be shown that the distribution of errors x about the true
value (taken as zero) is given by the law —

For, if d IS the amount of the perturbation, and positive and negative
perturbations are equally likely, the expected frequency of m positive
errors and n — m negative errors in N observations is the term
in iV'd+D", and the actual error is mS—{n~m)S—[2m^n)8, Similarly,
the frequency of the actual error -[2(^+1)—^]^^ is given by the term in
; and so on. Proceeding to the limit, as n becomes large,
we get the stated result precisely as for the limiting process of 8.15.

8.32 In the theory of errors it is more customary to write —

so that the distribution becomes —

y=^-h>x^ .... ( 8 . 22 )

^/7r

h is called the " precision (cf. 6.17). As h increases, the normal curve
becomes narrower and hence k measures in a sense the closeness of the
bulk of observations to the true value.

The occurrence of normal distributions in nature

8.33 It was found at an early date that error distributions followed
the normal law more or less closely, though it must be admitted not with
any great exactitude. The fact that many populations, particularly bio-
metrical populations such as those classified according to height and weight,
lie distributed round the mean in a humped curve which is not unlike the
normal curve, gave rise in the first half of the nineteenth century to keen
interest. Although the term normal had not then been applied, there
appears to have been a feeling that the curve was the ideal to which most
distributions should in some degree attain, and that an explanation was
demanded if they did not. The normal curve was, in fact, to the early
statisticians what the circle was to the Ptolemaic astronomers.

i88

THEORY OF STATISTICS

8.34 Workers during the latter half of the nineteenth century were
more careful not to let their theories outrun their facts, and as the data
accumulated it became evident that the normal distribution was no more
usual than any other type. In fact, rather the reverse, so that the occur-
rence of a normal distribution was to be regarded as something abnormal.

The reader may well ask,'' said Karl Pearson, '' is it not possible to find
material which obeys within probable limits the normal law ? I reply,
yes, but this law is not a universal law of nature. We must hunt for
cases."

The belief in the validity of the normal law in the theory of errors died
harder. " As M. Lippmann once said to me," says Poincar^, in his " Calcul
Probabilith” " Everybody believes in the law of errors, the experi-
menters because they think it is a mathematical theorem, the mathe-
maticians because they think it is an experimental fact."

8.35 One must, however, be careful not to go too far in seeking to avoid
an over-emphasis on the practical occurrence of the normal curve. A
certain number of distributions, more particularly those relating to
measurements on plants and animals, are approximately of the normal
form. As an example, we may take the distribution of Table 4.7, which
we show in fig. 8.3 fitted with a normal curve.

Place of the normal curve in theory

8.36 Strangely enough, the realisation that the normal distribution
did not correspond to any widespread natural effect did not diminish its
importance in statistical theory. On the con ti ary, the normal distribution
has increased in importance in recent years. It is instructive to consider
why this is so.

In the first place, the normal curve and the normal integral have
numerous mathematical properties which make them attractive and com-
paratively easy to manipulate. We have, for instance, already seen that
the moments and cumulants of the normal curve are expressible in simple
forms.

Now the normal form is reasonably close to many distributions of the
humped type. If, therefore, we are ignorant of the exact nature of a
humped distribution, or know the form but find it mathematically intract-
able, we may assume as a first approximation that the distribution is normal
and see where this assumption leads us. It is not infrequently found that
a population represented in this way is sufficiently accurately specified for
the purposes of the inquiry.

8.37 Secondly, we shall find, when we come to consider sampling
distributions, that many of the populations which occur are of the norm^
form; either exactly or to a satisfactory degree of approximation.

THREE THEORETICAL DISTRIBUTIONS

189

8.38 Thirdly, the theory of the normal curve has been applied to the
graduation of curves which are not normal.

Fig. 8.3. — The distribution of stature for adult males in the British Isles (fig. 4.6, page 83),

fitted with a normal curve

To avoid confusing the figure, the frequency-polygon has not been drawn in, the tops
of the ordinates being shown by small circles.

It is possible to develop a technique for expressing a given distribution

in the form of an infinite series whose terms depend on the quantity e 2
and certain dependent functions.

8.39 Fourthly, distributions which are not normal can sometimes be
brought to a form approximating to the normal by a transformation of
the variate. A population which is skew with respect to a variate x, for
instance, might be normal when we take as the variate. We gave an'
example of this kind of effect in Exercise 4.6, page 100, where we saw that a
population of men classified according to their weight was skew, whereas a
population classified according to height (which we may take to be roughly
proportional to the cube root of the weight) is nearly normal.

The Poisson distribution

8.40 We have found that the limit to the binomial would be a normal
curve even if and q were unequal, provided that n were increased sufficiently
to make {q--p) small compared with Vnpq. We now propose to iSud
the limit to the same series if one of the chances, say q, becomes indefinitely

190

THEORY OF STATISTICS

small and n is increased sufficiently to keep nq finite, but not necessarily
large — practical values are in fact usually small.

Let us suppose that q is very small and that qn is equal to the finite
number m.

In the binomial {p +qY, the term

qrpr^

r ! [n — r) !

n ! / inV / m\”

r 1 {n—r) !\« / \ n )

~Py~n) '

m\^

(8 23)

Now the limit of ^1 — — j as n becomes large

Applying Stirling's approximation (8.16) when n is large, the term

n !

(8.24)

V 27r[n — r)r^+^{w — r)^n'^ 1 —

-:r

Now the limit of

(-;)■

= ^, as we need not consider terms in which

f exceeds quantities of the order and the limits of ^1*—^^ ,

are both unity. Hence the limit of (8.24) is unity, and the limit of (8.23) is

nfer^

8.41 Hence the successive terms in the binomial are

and the limit of {q+PY

2 ! ^

g-w.

jn^

etc.

Wi+^+— +^+

(8.25)

THREE THEORETICAL DISTRIBUTIONS

I9I

This expression is called Poisson’s distribution, or Poisson’s exponential
limit It was first published by Poisson in 1837, but has subsequently
been rediscovered by numerous writers.

Constants of the Poisson distribution

8.42 Taking an origin located at the first term of the distribution, we
have —

0+m+(^x2)+(^x3)+ . .

/ ^ , m ,m^ , \

. . . 1

l+Y-j+^j+ . . . • • • 1

It may also be shown that —

' =m{m^ + 6^2 -\-7m + 1 )

From these results we have immediately —

M.eBiL=m ....
fi^:=:m{m+l) —nr

az=:Vm . , • . •

Hence,

a2=m=r=mean

(8.26)

(8.27)

8.43 The third and fourth moments about the mean will be found to be —

. • • . • (8.28)
.... (8.29)

so that

igs

THEORY OF STATISTICS

B L

^ /{j® w? m '

•

. (8.30)

•

. (8.31)

These results should be compared with the expressions

A-

npq

pqn

for the binomial. They are, as might be expected, the limits of those

expressions when q=^ and n is large.
n

8,44 We may state without proof that all the cumulants of the Poisson
distribution are equal to m»

THREE THEORETICAL DISTRIBUTIONS 193

8.45 Tables of the limit for various values of m and r have

r !

been published by several authorities. One such set will be found in
Tables for Statisticians and Biometricians, Part I.

The form of the frequency-polygon of the distribution (which, like the
binomial and unhke the normal, is discontinuous) can be judged from
fig. 8.4, in which the polygons for various values of in are drawn. It will
be seen that for low values of m the polygon is very skew, but that for
larger values it tends towards a symmetrical form.

8.46 The condition that p oi q shall be small, np or nq remaining finite,
implies that in practice we should expect to find a Poisson distribution
in cases where the chance of any individual being a success ** was small.
Such a case might arise, fot example, in considering the deaths from
a rare disease in a population, the chance of any individual dying from
it being small.

8.47 Attention to the fact that comparatively rare events are not
haphazard was first directed by Quetelet and von Bortkiewicz. The
latter's data of the number of men killed by the kick of a horse in certain
Prussian army corps in twenty years (1875-94) have become classical.

The frequency-distribution of the number of deaths in 10 corps per
army corps per annum over twenty years was —

Deaths Frequency

0 109

1 65

2 22

3 3

4 1

Here the total number of deaths was 122, and hence the mean deaths per
army corps per annum is 0*61. Taking this m, we find the following
values for various numbers of deaths per annum —

Frequency assigned by
Poisson's Limit

108-7
66-3
20-2
4-1

0-7 (4 and ovei)

If we calculate for the actual distribution, we find —

Deaths

0=0-78,

a2=0-6079

194

THEORY OE STATISTICS

Hence, is nearly equal to the mean, which is in accordance with theory.
The agreement is, in fact, very much closer than is usual. Many dis-
tributions are now available for the frequency of individuals who have met
with 0 , 1, 2 , . . . accidents, e.g. in factories, during a given period of time,
and more often than not such distributions give a value of the variance
exceeding the mean. This state of affairs can be accounted for on the
assumption that the individuals at risk have var 5 dng degrees of accident-
proneness,'' and the assumption has been corroborated by finding that
those individuals who have the largest number of accidents in one period
are, on the whole, those who have most accidents during a succeeding period.

A more modern example of the occurrence of the distribution is given
in the following data relating to the incidence of fi 3 dng bombs (VI) in an
area in south London. An area of 144 square kilometers was selected
for which the mean density of bombs appeared constant. To test the
hypothesis that the bombs fell in clusters the area was divided into 576
squares of J kilometer each and a count made of the numbers of squares
containing 0, 1, 2, etc. bombs, of which there were 537 altogether. A
comparison with the frequencies given by a Poisson distribution is as
follows (data from R. D. Clarke, 1948, Jour, Inst, Act., 72, No. 335) —

Number of flying
bombs pet square

Actual
number of
squares

Theoretical number
given by the
Poisson distribution

229

226-74

211

211-39

98-54

30-62

7-14

5 and over

1-57

Total

576-00

The agreement is extraordinarily close and there appears no evidence
that the bombs clustered " otherwise than by chance.

It is an interesting reflection that although the cavalry of 1875 developed
into the flying bomb of 1945 the laws of probability seem to have endured
over this span of 70 years.

Another example of the Poisson distribution is given in Exercise 8.17
at the end of this chapter. The early instances of the distribution were
nearly all demographic, and for some time it remained more of a curiosity
than a useful tool. In 1907, however, '' Student " drew attention to a
class of hsemacytometer counts to which the distribution seemed appropri-
ate, and since that time it has found several important biological applica-
tions. It also appears in problems of controlling road and telephone traffic.

Pearson curves

848 The process of obtaining the normal curve as a limit of the binomial
suggested to Karl Pearson an investigation into a series of analogous

THREE THEORETtCAL iDlStRtBtJtlORS

195

curves which may be regarded as limits to skew binomials or to distributions
from a finite population, e.g. by drawing r balls at a time from a bag which
contains a finite number iV of black and white balls in given proportions.
One such curve was of the form

This set of curves, divided into twelve types, w^hich were later regarded
from rather a different standpoint, can be made to fit a large number of the
distributions occurring in practice.

In the curve given above, y, a and the origin can all be obtained from
the first three moments. For the other curves of Pearson's system,
except some degenerate types, the first four moments are necessary to
specify the constants of the curve completely. The distributions con-
sidered hitherto have required in addition to the area (number of observa-
tions), either the mean only (Poisson) or the mean and standard deviation
(normal curve) to determine their constants ; but the principle of fitting
for the more general curves remains the same. The actual moments of
the curves are equated to the moments expressed in terms of the constants,
such as y and a, which are to be found. For full details of these curves,
the method of determining the type to choose and the method of fitting,
the student is referred to Elderton's Frequency Curves and Correlation
and Kendall's Advanced Theory of Statistics, vol. i.

SUMMARY

1 . If the chance of the success of an event is and of its failure q, then,
provided that the chance remains constant throughout the trials, the
expected frequencies of 0, 1,2,... successes in N sets of n trials are the
1st, 2nd, etc. terms in the binomial

N{q+py

2. The mean of the binomial is pn and its standard deviation is V npq,

3. For the binomial —

npq

A=34

pqn

4. If neither p nor q is small, the binomial tends for large values of n
to the form

196

THEORY OF STATISTICS

2or»

5. This curve, which may also be written

N -J

is called the normal curve.

6. The standard deviation of the normal curve is o*. Its third moment
is zero, and the fourth moment is 3a^. Hence

^ 2=3

All cumulants higher than the second are zero.

7. In the theory of errors the normal population is usually written —

being called the precision.

8, The mean deviation of the normal curve is

a^=0- 79788 ... a

and the quartile deviation (semi-interquartile range) is 0 • 67448975 . . , <j

9. A range 3a on each side of the mean of the normal curve contains
0-9973 of the distribution.

10. If ^ or y is small and one of pn, qn is finite and equal to m, the
binomial distribution tends to the limit

(r«n+»t+— + . . . +— + • • • }

This is called the Poisson distribution.

11 . The mean of the Poisson distribution is m, and a^ also equals m,

12. For the Poisson distribution —

and all the cumulants are equal to m.

EXERCISES

8.1 A perfect cubic die is thrown a large number of times in sets of 8.
The occurrence of a 5 or a 6 is called a success. In what proportion of the
sets would you expect 8 successes ?

THREE THEORETICAL DISTRIBUTIONS

197

8,2 The following data, due to W. F. R. Weldon, show the results of
throwing 12 dice 4,096 times, a throw of 4, 5 or 6 being called a success —

Successes

Frequency

Successes

Frequency

—

847

536

257

198

430

731

—

948

Total

4,096

Find the expected frequencies, and compare the actual mean and standard
deviation with those of the expected distribution.

8.3 In the previous example find the equation of the normal curve which
has the same mean, standard deviation and total frequency as the observed
distribution.

Find the frequencies to be expected if the distribution were represented
exactly by the ordinates of this curve and compare them with the actual
frequencies.

8.4 Assuming that half the population are consumers of chocolate, so that
the chance of an individual being a consumer is and assuming that 100
investigators each take ten individuals to see whether they are consumers,
how many investigators would you expect to report that three people
or less were consumers ?

8.5 An irregular six-faced die is thrown, and the expectation that in 10
throws it will give five even numbers is twice the expectation that it will
give four even numbers. How many times in 10,000 sets of 10 throws
would you expect it to give no even numbers ?

8.6 If two normal populations have the same total frequency but the a
of one is k times that of the other, show that the maximum frequency of

the first is -=■ that of the other.
k

8.7 Find graphically or otherwise the point of inflection of the normal
cu* , o, and show that it occurs at a distance a from the mean ordinate.

8.8 Show that if np be a whole number, the mean of the binomial coincides
with the greatest term.

8.9 Show that if two symmetrical binomial distributions of degree %
(and of the same number of observations) are so superposed that the rth
term of the one coincides with the (^^+l)th term of the other, the distribu-
tion formed by adding superposed terms is a symmetrical binomial of
degree (w+1).

[Note , — It follow^ that if two normal distributions of the same area and
standard deviation are superposed so that the dijSerence between the

198

THEORY OF STATISTICS

means is small compared with the standard deviation, the compound
curve is very nearly normal.]

8.10 Calculate the ordinates of the binomial 1,024 (0*5 +0*5)^® and
compare them with those of the normal curve.

8.11 If skulls are classified as dolichocephalic when the length-breadth
index is under 75, mesocephalic when the same index lies between 75 and 80,
and hr achy cephalic when the index is over 80, find approximately (assuming
that the distribution is normal) the mean and standard deviation of a
series in which 58 per cent are stated to be dolichocephalic, 38 per cent
mesocephalic and 4 per cent brachycephahc.

8.12 Find the deciles of the normal curve.

8.13 Write down the normal population which has the same mean and
(uncorrected) standard deviation as that of the last column of Table 4.7,
page 82, and find the mean deviation and quartile deviation. Compare
the results with the corresponding quantities for the actual distribution.

8.14 Proceed similarly for the skew population of Table 4.8, page 84.

8.15 In Exercise 10.4, if 1,000 investigators each choose 100 individuals,
how many would you expect to report that more than 60 persons are
consumers ?

8. 16 Taking the population of screws of Table 4.3, page 72, find the normal
population which has the same standard deviation and a mean of 1 inch.
Compare the frequencies given by this population with the actual
frequencies.

8.17 The following data (Lucy Whitaker, Biometrika, 1914, 10, 36) give
the number of deaths of w^omen over 85 published in The Times during
1910-12—

Number of deaths
per day

6
7

Frequency

364

376

218

Find the frequencies of the Poisson distribution which has the same mean
as this distribution, and compare your results with the actual frequencies.
For the purpose of this example, simple interpolation in the tables given
in Tables for Statisticians and Biomeiricians is sufficient.

8-18 In the data of the previous exercise calculate the first four
cumulants.

CHAPTER NINE

CORRELATION AND REGRESSION

Bivariate populations

9.1 In Chapters 4 to 8 we considered the members of a population
classified according to the values of a single variable ; and we saw how
they could be grouped into a frequency-distribution whose character-
istics could be described by certain constants. We have now to proceed
to the case of two variables, in which each member of the population will
exhibit two values, one for each of the variables under consideration.

A population of this kind is called a bivariate population. One of our
main topics will be the way in which the two variables are related in the
population.

9.2 If the corresponding values of the two variables are noted for each
member, the methods of classification employed in the previous chapters
may be applied to both variables. We can thus group our data into a
table of double entry, or contingency table (Chapter 3), showing the
frequencies of pairs of values l3dng wdthin given class-intervals. Six
such tables are given below as illustrations for the following variables :
Table 9.1, two measurements on a shell; Table 9.2, ages of husbands
and their wives in marriages taking place in England and Wales in 1933 ;
Table 9.3, statures of fathers and their sons ; Table 9.4, age and yield of
milk in cows ; Table 9.5, the rate of discount and ratio of reserves to
deposits in American banks; Table 9.6, the birth rate per thousand and
the total numbers of births in the registration districts of England in
1941.

Arrays and correlation tables

9.3 Each row in such a table gives the frequency-distribution of the
first variable for the members of the population in which the second variable
lies within the limits stated on the left of the row. Similarly for the
columns. As ** columns and “ rows " are distinguished only by the
accidental circumstances of the one set running vertically and the other
horizontally, and the difference has no statistical significance, the word
array has been suggested as a convenient term to denote either a row or
a column.

If the values of X in one array are associated with values of Y in an
interval centred at Yu, then Yn is called the type of the array.

199

200

THEORY OF STATISTICS

CORRELATION AND REGRESSION

201

9.4 A grouped frequency-distribution of the type of Tables 9.1 to
9.6 may then be termed a bivariate frequency-distribution ; but if we are
particularly interested in the relationship between the two variates it is
sometimes called a correlation table. The difference between a correlation
table and a contingency table lies in the fact that the latter term may
be, and usually is, applied to tables classified according to unmeasured
quantities or imperfectly defined intervals.

9.5 We need add very little to what was said in Chapter 4 about the
choice and magnitude of class-intervals and the classification of data.
When the intervals have been fixed, the table is readily compiled from the
raw material by taking a large sheet of paper ruled with arrays properly

TABLE 9.2 — Correlation between ages of (1) husband and (2) wife in marriages in
England and Wales in 1933

Figures in hundreds— certain marriages in which no age was specified are omitted.
(Data from Registrar-General’s Statistical Review of England and Wales for 1933, Tables, Part 11, Civil)

(2) Age of
wife

15-

20—

25-

(1) Age of husband (Years)

30^ 35- 40- 45- 50- 55- 60- 65- 70- 75-

Total

(Years)

15-

189

288

20-

682

585

106

—

1,418

25-

140

511

179

—

896

30-

—

101

—

268

35-

—

112

40-

—

45-

—

50-

—

55-

—

60-

—

65-

—

70-

—

' —

—

Total

1,024

1,238 424

143

3,153

headed in the same way as the final table and entering a small mark in
the compartment corresponding to the variate values exhibited by each
individual. If facility of checking be of great importance, each pair of
recoided values may be entered on a separate card and these dedt into
little packs on a board ruled in squares, or into a divided tray ; each pack
can then be run through to see that no card has been mis-sorted. The
difificulty as to the intermediate observations — ^values of the variables
corresponding to divisions between class-intervals — will be met in th6 same
way as before if the value of one variable alone be intermediate, the unit
of frequency being divided between two adjacent compartments. If both
values of the pair be intermediates, the observation must be divided
between fom adjacent compartments, and thus quarters as well as halves

TABLE 9.3 — Correlation between (1) stature of father and (2) stature of son : 1 or 2 sons only of each father

202 THEORY OF STATISTICS

CORRELATION AND REGRESSION

203

may occur in the table, as for example, in Table 9.3. In this case the
statures of fathers and sons were measured to the nearest quarter-inch
and subsequently grouped by 1-inch intervals : a pair in which the recorded
stature of the father is 60*5 in. and that of the son 62-5 in. is accordingly
entered as 0*25 to each of the four compartments under the columns
59-5-60*5, 60 *5-^1 *5, and the rows 61 *5-^2 *5, 62 *5-63 -5.

Frequency-surface and stereogram

9.6 The distribution of frequency for two variables may be represented
by a surface in three dimensions in the same way as the frequency-
distribution for a single variable may be represented by a curve in two.
We may imagine the surface to be obtained by erecting at the centre of
every compartment of the correlation table a vertical of length proportion-
ate to the frequency in that compartment, and joining up the tops of the
verticals. If the compartments were made smaller and smaller while the
class-frequencies remained finite, the irregular figure so obtained would
approximate more and more closely towards a continuous curved surface
— a frequency-surface — corresponding to the frequency-curves for single
variables of Chapter 4. The volume of the frequency-solid over any area
drawn on its base gives the frequency of pairs of values failing within that
area, just as the area of the frequency-curve over an interval of the base
line gives the frequency of observations within that interval.

9.7 Similarly, a figure analogous to the frequency-polygon or the
histogram may be constructed by drawing the frequency-distributions for
all arrays of the one variable, to the same scale, on sheets of cardboard,
cutting-out and erecting the cards vertically on a base-board at equal
distances apart, or by marking out a base-board in squares corresponding
to the compartments of the correlation table, and erecting on each square
a rod of wood of height proportionate to the frequency. Such solid repre-
sentations of frequency-distributions for two variables are sometimes
termed stereograms.

9.8 It is impossible, however, to group the majority of frequency-
surfaces, in the same way as the frequency-curves, under a few simple
types : the forms are too varied. The simplest ideal type is one in which
every section of the surface is \ symmetrical curve — the first type of
Chapter 4, fig. 4.5, page 81. Like the symmetrical distribution for the
single variable, this is a very rare form of distribution in economic statistics,
but approximate illustrations may be drawn from anthropometry. Fig.
9.1 shows the ideal form of the surface, somewhat truncated, and fig. 9,3
the distribution of Table 9.3, which approximates to the same type —
the difference in steepness is, of course, merely a matter of scale. The
maximum frequency occurs in the centre of the whole distribution, and
the surface is symmetrical round the vertical through the maximum, equal
frequencies occurring at equal distances from the mode on apposite sides.

TABLE 9.4— Correlatioii between (1) age in years and (2) yield of milk per week In 4,912 Ayrshire co\^
(Data from J. F. Todier, “ An Investigation of the Milk Yield of Dairy Coira,” Bumetnkat 1928, 20B, 106)

204

THEORY OF STATISTICS

Totals 112 1,129 1,047 812 636 419 276 223 122 75 32 15 7 2 4 1 4,912

CORRELATION AND REGRESSION

205

2o6 theory of statistics

CORRELATION AND REGRESSION

207

TABLE 9.7 — Showing the monthly index-numbers of prices of (1) animal feeding-stuffe
and (2) home-grown oats in England and Wales for 1931-1935
The index-numbers are based on prices m Cf^’Te‘?"^or:dine months of 1911-1913
(Data from Agncuitural Market Repo-i ■ ! 'id .md Wales)

Month

Index of
feeding-stuffs
price

Index of
oats
price

Month

Index of
feeding-stuffs
price

Index of
oats
pnce

1931 Jan.

1933 July

Feb,

Aug.

Mar,

Sept.

Apr.

Oct.

May

Nov.

June

Dec.

Jiiiy

Aug.

1934 Jan.

Sept.

Feb.

Oct.

Mar.

Nov.

Apr.

Dec.

May

June

1932 Jan.

102

July

Feb.

102

Aug,

101

Mar.

102

105

Sept.

102

Apr.

105

Oct.

May

107

Nov.

June

107

Dec.

July

101

Aug.

106

1935 Jan.

100

Sept.

Feb.

Oct.

Mar.

Nov.

Apr.

i 90

Dec.

May

! 88

June

1 86

1933 Jan.

July

^ 83 i

Feb.

Aug.

1 80

Mar.

Sept.

L 81

Apr.

Oct.

May

Nov.

June 1

Dec.

The next simplest type of surface corresponds to the second type of
frequency-curve — the moderately asjmmetrical. Most, if not all, of the
distributions of arrays are asymmetrical and like the distributions of fig.
4.7 ; the surface is consequently asymmetrical, and the maximum does
not lie in the centre of the distribution. This form is fairly common, and
illustrations might be drawn from a variety of sources — economics,
meteorology, anthropometry, etc. The data of Table 9.4 will serve as an
example. The total distributions and the distributions of the majority
of the arrays are asymmetrical, the rows being markedly so. The maximum
frequency lies towards the upper end of the table in the compartment
under the row headed ‘‘ 16 ” and column headed 4 The frequency
fails off very rapidly towards the lower ages, and slowly in the direction
of old age.

Apart from these two forms, it seems impossible to delimit empirically
any simple types. Tables 9.5 and 9.6 are given simply as illustrations of

2o8

THEORY OF STATISTICS

two very divergent forms. Fig. 9.2 gives a graphical representation of the
former by the method con'esponding to the histogram of Chapter 4, the
frequency in each compartment being represented by a square pillar. The
distribution of frequency is very characteristic, and quite different from
that of any of the Tables 9.1 to 9.4.

mg. 9.2.— Freanency-sutfece for tte rate of dleconnt and ratio of reserve*

CORRELATION AND REGRESSION

2II

The scatter diagram

9.9 There is another method of representing bivariate data graphically
which is particularly useful for ungrouped data. Take, for instance,
the data of Table 9,7, giving the index-numbers of prices of animal feeding-
stuhs and home-grown oats for each month of the years 1931-35. There
are only 60 pairs of values, and the data cannot be grouped into a
frequency-distribution with class-intervals of reasonable size without

Index lumber of feeding-staffs prices

Fig. 9.4. — Scatter diagram of index-numbers of prices of (1} animal feeding-stuffs and
(2) home-grown oats (Table 9.7)

For the meaning of the straight lines, see Example 9.1, page 223

giving rise to irregular frequencies. We may, however, proceed as
follows —

On squared paper take two axes at right angles, one axis corresponding
to the variable X and the other to the variable Y (see fig. 9.4). To each
member of the population there will correspond a pair of values X, Y, which
in turn will correspond to a point whose abscissa on the diagram is X and

212

THEORY OF STATISTICS

v/hose ordinate is Y. Thus the population, when represented in this way,
will give a swarm of points on the diagram, and we can interpret the ways
in which these points cluster or scatter as properties of the relationship
between the two variables. Fig 9.4 shows the data of Table 9.7 plotted
in this way. It will be observed that the points tend to distribute them-
selves so that high and low’ values of X correspond to high and low values
of Y respectively.

Such a figure is called a scatter diagram,

9.10 We can also represent a grouped bivariate frequency table on
a scatter diagram, though less satisfactorily and with some labour. For
this purpose axes are taken as before and abscissas and ordinates drawn to
correspond to the divisions of the frequency table. The diagram wiU then
be divided into compartments corresponding to the compartments of the
table. In each compartment we place a number of dots equal to the
frequency in the corresponding compartment of the table. We have, as a
rule, no guide as to the disposition of these dots within their respective
cells, and hence it is usual to place them in some symmetrical arrangement
so that they are, as nearly as may be, spread uniformly through the cells.

The difficulty of inserting the dots when the frequencies are large will
be obvious, and, in fact, such a scatter diagram rarely tells us more than we
can see from an inspection of the table itself. In contrast to this, the
scatter diagram of the data of Table 9.7 gives a much better picture of the
dependence of the two variates than can be obtained by mere inspection
of the ungrouped data of the table.

9.11 It is clear that a correlation table may be treated by the methods
discussed in Chapter 3, which are applicable to all contingency tables,
however formed. But the coefficient of contingency merely tells us
whether two yariables are related, and if so, how closely. The methods
we shall now discuss go much further than this. The numerical character
of the variates and the arrangement of the correlation table in class-
intervals of equal widths enable us to approach the problem of investigat-
ing the relationship between the variates with additional precision.

9.12 If the two variates in a contingency table are independent, the
distributions in parallel arrays are similar (3.18) ; hence their averages
and dispersions, i.e. their means and standard deviations, must be the same.
In general they will not be the same, and we are thus led to inquire into the
relation between the values of the means and standard deviations in
different arrays and the departure of the distribution from complete
independence,

9.13 The mean is the most important constant, in general, and for
the present we shall concentrate our attention upon it. Although the
values in arrays are scattered about their respective means, it is in most
cases profitable to inquire how the means of arrays are related ; this will

CORRELATION AND REGRESSION

213

throw a good deal of light on the important question whether high values
of one variate show any tendency to be associated, on the average, with high
values of the other variate.

If possible, we also wish to know how great a divergence of one variate
from its mean is associated with a given divergence of the other, and to
obtain some idea of how closely the relation is usually fulfilled .

Lines of regression

9.14 Let us then consider the means of arrays. Let OX, OY he two
axes at right angles representing the scales of the two variates. As in
the case of the scatter diagram we can plot the positions of the means ; for
example, if the mean of a row whose variate value is centred at is

we can plot the point whose abscissa is and whose ordinate is y^. There
will thus be one point corresponding to each row and one to each column.
In practice, to distinguish the two, the means of rows are denoted by small
circles and the means of columns by small crosses. Fig. 9.8 shows such
a diagram drawn for the data of Table 9.3.

The means of rows and the means of columns will, in general, lie more
or less closely round smooth curves.^ For example, in fig. 9.8 they lie,
very approximately, on straight lines, RR and CC in the figure. Such
curves are said to be curves of regression, and their equations with reference
to the axes OX and OY are called regression equations. If the lines of
regression are straight, the regression is said to be linear. In the contrary
case it is said to be curvihnear.

9.15 The term regression is not a particularly happy one from
the etymological point of view% but it is so finiily embedded in statistical
literature that w^e make no attempt to replace it by an expression which
would more suitably express its essential properties. It was introduced by
Galton in connection with the inheritance of stature. Galton found that
the sons of fathers who deviate x inches from the mean height or all fathers
themselves deviate from the mean height of all sons by less than x inches,
i.e. there is what Galton called a ** regression to mediocrity.'* In general
the idea ordinarily attached to the word regression " does not touch
upon this connotation, and it should be regarded merely as a convenient
term.

9.16 If two variates are indepe ndent, their regression lin es ar e straight
2 tnd at right ahS^^ ttiT^meaS^of rows lying on a TS y^SarSferto the

on a line paraUeMo Jjhe

ToFtl^dStribi^ions i n pa rallel ar rays are"siSnIaf"^see fig. 9.5). In a nv
case’drawn from actual data, of cdufseTTHTm^ns might not lie exactly on
straight lines, owing to fluctuations of satnipiing,

9.17 The cases with which the experimentalist, e.g. the chemist or
physicist, has to deal, where the observations are all crowded closely
round a single line, lie at the opposite extreme from independence. The

214

tfiEORV OE STATISTICS

entries fall into a few compartments only of each array, and the means of
rows and of columns lie approximately on one and the same curve, like
the line RR of fig. 9.6.

9*18 The ordinary cases of statistics are intermediate between these
two extremes, the lines of means being neither perpendicular as in fig. 9.5,
nor coincident as in fig. 9.6. One problem of the statistician is to find
expressions which vdJi suffice to describe the regression lines, either exactly
or to a satisfactory degree of approximation.

In general this is a difficult problem, and the theory of curvilinear
regression is as yet incomplete. We can, however, make considerable
progress by confining ourselves to the cases in which the regression is linear.
Cases of this kind are more frequent than might be supposed, and in other
cases the means of arrays lie so irregularly, owing to the paucity of the
observations, that the real nature of the regression curve is not indicated
and a straight line will give as good an approximation as a more elaborate
curve,

9.19 Consider the simplest case in* which the means of rows lie exactly
on a straight line RR (fig. 9.7). Let Mg be the mean value of Y, and
let RR cut M 2 X, the horizontal through Mg, in M. Then it may be
shown that the vertical through M must cut OX in M^, the mean of X,
For, let the slope of RR to the vertical, i.e. the tangent of the angle M^MR
or ratio of kl to IM, be and let deviations from My, Mx be denoted by x
and y.

00 7 2 4. B BX 0 7 2 sm 4 B 6

Fig. 9.5

Fig. 9.6

CORRELATION AND REGRESSION

215

Fig. 9.7

Then for any one row of type y in which the number of observations
is n, 'L{x)—nbjy. and therefore for the whole table, since S(wy)=0,
S(a:)=6iS(M>')=0. Ml must therefore be the mean of X, and M may
accordingly be termed the mean of the whole distribution.

Knowing that RR passes through the mean of the distribution, w’e can
determine it completely if we know the value of bi

For any one row we have

'Z[xy)—yZ{x) =n\y^

Therefore for the whole table

Jl{xy) —b^{y^)n—Nbia/

Let us write

2i6

THEORY OF STATISTICS

Now let US define

_ ^(xy)

• ( 9 - 4 )

(JxCSy

Then

&i=r— and 6o=r- .... (9.5)

and the equations of RR and CC, referred to the centre of the distribution,
are

(Jyf

Xz=:zY~.y

(Jy

and, referred to the origin 0,

and y=r^x .
ax

. (9.6)

(9.7)

9.20 Let us now proceed to the case when the means of arrays are not
situated on a straight line. This we shall treat by finding the next best
thing — straight lines which are the closest fit to the means.

The expression closest fit/' as applied to the fitting of curves to points,
is one which we deal with at length in Chapter 15, and it is only necessary
to say at this stage that the straight line RR of closest fit to the means of
rows, i,e.

will be determined by evaluating and so as to make the expression

E^^x^{a^+b,y)y

(that is, the sum of the squares of the horizontal distances of the points
representing the observations from RR) a minimum. Here x and y,
as before, denote deviations from the xespective means of X and Y, and
the summation is taken over all values of x and y .

We have, expanding E,

E -Z ^b^y) } +X{x ^b^y) ^

The second term on the right vanishes, since S(^)=Z( y)=0 and hence

Now and 6^ can be chosen independently, and hence £ is a mininium
only if 0, i.e.

(9.8)

Thus the line of closest fit goes through the mean of the distribution.

CORRELATION AND REGRESSION

217

Hence*

=-'z\x^) -2b^{xy) +5i2S( v 2 )
^{xy)Mx^)

=2(3'^) j6x*-26i;

=2M[{

2(3'^)

'^Y

S(3'^)i

S(v^)

Hiy^)

(9.9)

This is a minimum when the first term (a square) is zero. i.e. when

_^{xy]
2(3'*) ■

. (9.10)

which is the same as equation (9.2).

We may show similarly that the line of closest fit CC, given by

has

^ 2 —

which is the same as equation (9.3).
If we regard the equation

x-^a^+b^y

as one for estimating x from y, we may take x-^Uj as the error of
estimation, and E will then be the sum of the squares of such errors. The
condition that E is a minimum is then equivalent to the condition that the
sum of squares of errors of estimation shall be a minimum. This is one
form of the so-called '' Principle of Least Squares (see Chapter 15).

9.21 Equations (9.6) and (9.7) are thus of general application. If the
regression is exactly linear they give the lines of regression. If the
regression departs from linearity, either owing to sampling ef ects or owing
to real divergences, they give the best '' straight regression lines which
the data admit. We may regard the equations as either {a) equations for
estimating an individual % from its associated y (ory from its associated x)
in such a way that the sum of squares of errors of estimation is a minimum ;
or (b) equations for estimating the mean of the x*s associated with a
particular y (or the mean of y's associated with a particular x) in such, a
way that the sum of the squares of errors of estimation is a minimum,
each mean being counted proportionately to the number of observations
on which it is based.

2I8

THEORY OF STATISTICS

Coefficient of correlation

9.22 The coefficient r defined in equation (9.4) is of very great importance.
It is called the coefficient of correlation,
r cannot exceed +1 or be less than —1.

For, from equation (9.9) we see that the value of E is

. . (9.11)

But E is the sum of a number of squares and cannot be negative.
Hence,

which proves the result.

If r=:4-i, the regression equations are identical, as may be seen from
equations (9.6), and hence the lines RR and CC coincide. In this case it
follows from (9.11) that for all pairs of values of the variates

x—biy==^0

i.e. all values lie on a single straight line. Thus to one value of x there

JTalher stotlure^

Fig. 9,8. — Correlation between stature of father and stature of son (Table 9.3)
Means of rows shown by circles and means of columns by crosses : 4-0 '51

CORRELATION AND REGRESSION

219

corresponds one, and only one, value of y. This is the case we mentioned
in 9.17, and since high values of x correspond to high values of y, the
variables may be said to be perfectly positively correlated.

Similarly, if r =— 1, the pairs of values all lie on a single straight line as
before, but high values of one will be associated with low values of the

Jl^e , iiL years

Fig. 9.9. — Correlation between age and weekly yield of milk from cows (Table 9.4)
Means of rows shown by circles and means of columns by crosses • r — +0*22

other. In this case we can say that the variates are perfectly negatively
correlated.

Finally, if the variates are independent, r is zero, for and h^, are zero,
and the lines of regression are parallel to OX and OY , It does not follow,
however, that if r is zero the variates are independent ; the fact that r is
zero implies only that the means of arrays lie scattered around two straight
lines which do not exhibit any definite trend away from the horizontal or
the vertical as the case may be. Two variates for ^which r is zero may,
however, be spoken of as uncorrelated. Table 9,6 will serve as a case
where the variates are almost uncorrelated but by no means independent,

Births

Fig. 9.]

Means c

CORRELATION AND REGRESSION

221

f being small (0-17) (see fig. 9.10), but the coefi&cient of contingency C
(for the grouping of Exercise 9.3) 0*30. Figs. 9.8 and 9.9 are drawn from
the data of Tables 9.3 and 9.4, for which r has the values 4-0*51 and
4-0*22 respectively. The student should study such tables and diagrams
closely, and endeavour to accustom himself to estimating the value of r
from the general appearance of the table.

It does not follow that if % andy are functionally related their correlation
is unity, unless the relationship is linear. Cf Exercise 10.9.

Coeffidente of reagression
9.23 The two quantities

are called coefficients of regression, being the regression of x on y, or
deviation in x corresponding on the average to a unit change in y, and
being similarly the regression of y on x.

The coefficient of correlation is always a pure number, but the coefficients
of regression are only pure numbers if the variates are the same in kind ;

for they depend, on the ratio — , and consequently on the units in which

<Jy

X and y are measured.

Since r is not greater than unity, one of the coefficients of regression is

less than unity ; but the other may be greater than unity, if — or — be
large.

9.24 The two standard deviations,

Sx=ax\^l--r^, Sy=ayVl —

are of considerable importance. It follows from (9.11) that Sx is the
standard deviation of [x—h^y), and similarly Sy is the standard deviation
of {y—h^). Hence we may regard Sx and Sy as the standard errors (root-
mean-square errors) made in estimating x from y and y from x by the
respective regression equations

Sx may also be regarded as a kind of average standard deviation of a row
about RRy and Sy as an average standard deviation of a column about CC.
In an ideal case, where the regression is truly linear and the standard
deviations of all parallel arrays are equal, a case to which the distribution
of Table 9.3 is a rough approximation,^ Sx is the standard deviation of the
itr-array and Sy the standard deviation of the y-array. Hence Sx and Sy are
sometimes termed the standard deviations of arrays.**

^ Tables in wbich the standard deviations of arrays are equal are sometimes said
to be “ homoscedastic ; in the contrary case " heteroscedastic.*’

222

THEORY OF STATISTICS

Calculation of the coefficient of correlation

9.25 We now proceed to the arithmetical work involved in calculating
the correlation coefficient.

For this purpose we use the formula (9,4), i.e.

l,{xy) i:{xy)

y/'Z{x^yZ{y^)

The calculation of or and of S(y2), or Oy, proceeds exactly

as in Chapter 6. The only expression of a novel type is the quantity

which we may call the first product-moment or the covariance

of the distribution.^ As in the case of univariate distributions, the form
of the arithmetic is slightly different according as the observations are
grouped or ungrouped.

9.26 Our work is greatly simplified by the use of devices similar to those
employed in calculating the means and other moments of univariate
distributions,

(a) We take working means for the two variates, obtained by inspection,
and tiansfer our moments to those about the means after the bulk of
the arithmetic has been performed. For the first product-moment we
have, in fact, if tj are the deviations from the working means and
^ the deviations of the true means from the working means —

v^y+v

Hence,

Summing for all members of the population, since S(|y) =0 and

similarly =0, x and y being deviations from the true means,

Hence,

Jl{xy)=-S{iv)-Nlv .... (9.12)

This gives us the product-moment about the true means in terms of
the product-moment about the working means and the deviations of the
true means from the working means.

?• In generalisation of the definition of moments of a univariate distribution in
Chapter 7 we may define the product-moments of a bivariate population as

where / is the frequency and the variates are measured from their means. This gives us

/‘u==^S{/*y)

the quantity we have called p in equation (9.1).

CORRELATION AND REGRESSION

223

(J) As a check on the rather heavy arithmetic which is frequently
involved, it is advisable to use a method similar to that of 6.11. We have

S(^+l){^ + l)=2(g^)+S(g)+S(7/)+iV . . . (9.13)

If, therefore, we calculate S(g+l)(i 7 -f-l) as well as S(^ 7 ), we shall have in
the above equation a check on the accuracy of our work.

Example 9.1, Table 9.8. — Let us investigate the correlation and re-
gressions of the variates of Table 9.7, the data of which are ungrouped.
The variates are (1) the price index-number of animal feeding-stuffs, X,
and (2) the price index-number of home-grown oats, Y, The values of
the variates themselves are shown in columns 2 and 3 of Table 9,8. We
take a working mean at X =90 and Y =90, and the deviations from these
values are shown in columns 4 and 5. The remaining columns 6 to 13
give the squares and product of the deviations together with the various
auxiliary quantities used for checking purposes. Finally, the various
sums are shown at the bottom of the table.

In practice it is as well to show the negative values which may occur in
columns 4, 5, 6, 7, 12 and 13 (particularly the last two) in a separate column,
so as to facilitate addition and avoid mistakes. We have refrained from
this course for convenience of printing.

As check on the arithmetic we have —

-1 18=S(g) ==S(§-+-1) ~iV= -58 -60

2,924=S(^+l)2=S(g2)+2S(g)-)-iV=3,100-236-feO

etc., and

2,493=S(^+1)(9?+1) =2(^9;) +i:(^) -fS(9/) -fiV'
=2,565-118-14 +60
=2,493

We have, then, about the working means —

5— g 0-2333

ct, 2=?^-|«=47-7989, a,=6*914

a,*='^^-^j2=80-1789, cr5,=8-954

^=5M=5M_|^=42.75 -0-4589 =42 -2911

P 42-2911 . „

'' “61 ■ 9080 -

224

THEORY OF STATISTICS

TABLE 9.8 — Correlation between monthly index-numbers of prices of (1) animal
feeding-stuffs and (2) home-grown oats in years 1931-35

CORRELATION AND REGRESSION 225

Further, working the regressions in the way best to avoid errors in
rounding off,

=0-527

=-.=0-885

Thus the correlation coefficient is 0-68, and the regression equations,
referred to the means, are —

x=Q^S27y

y^O^SSSx

If we prefer to express these equations with origin at X = 0, ¥=0,
we have —

X - (90 -- 1 • 97) • 03 =0 • 527(y -89 • 77)

Y-(90-0*23)=Y-89*77 =0-885(X-88*03)

which reduce to

Z-0-527Y+40-72 . ... (a)

Y=0-885X+ll-86 . ... (b)

The lines of regression are drawn on the scatter diagram of fig. 9.4,
The standard errors made in using these equations to estimate the
index-number of oats from animal feeding-stuffs, and vice versa, are —

a4;Vr-r2=5-07

ayVl-r2==6*57

Equation (a) tells us that a rise of one point in the price index-number of
oats is accompanied on the average by a rise of 0*527 point in the price
index-number of feeding stuffs. Similarly, equation (b) tells us that a
rise of one point in the index for feeding-stuffs is accompanied average
by a rise of 0*885 point in the pi;ice of oats.

It is important to note that the regression equations do not tell us
whether a variation in one variate is caused by a variation in the other ;
aJl we know is that the two vary together, and so far as the regression
equations show, either the feeding-stuffs price may exert an influence on
the oats price, or vice versa, or their common variation may be due to
some other cause affecting both. This is only one instance of a difficulty
which pervades the theory of correlation and regression, namely, that
of interpreting results in terms of causal factors.

TABLE 9.9. — Coirrelation between (1) length of moth^-frond, (2) length of daughter-frond, in Lemna minor

(The frequencies are the figures printed in ordinary type. The numbers in heavy type are the deviation-products (^17) )

(Unpublished data , G. U. Yule)

226

THEORY OF STATISTICS

CORREtAttON AND REGRESSION

Example 9.2, Table 9.9. — We now consider an example based on
grouped data. In this w^e have omitted the auxiliary quantities necessary
for checking in order to save space.

(Unpublished data ; measurements by G. U. Yule.) The two variables
are (1) X, the length of a mother-frond of duckweed [Lemna minor) ;
(2) Y , the length of the daughter-frond. The mother-frond was measured
when the daughter-frond separated from it, and the daughter-frond when
its first daughter-frond separated. Measures w’'ere taken from camera
drawings made with the Zeiss- Abbe camera under a low power, the actual
magnification being 24 : 1. The units of length in the tabulated measure-
ments are millimetres on the drawings.

The arbitrary origin for both X and Y was taken at 105 mm. The
following are the values found for the constants of the single distributions —

^ — 1 * 058 intervals = ~ 6 • 3mm. = 98*7 mm . on drawing

= 4*11 mm. actual

av= 2*828 intervals = 17*0 mm, on drawing == 0*707 mm. actual

= — 0*203 interval^— 1 *2 mm. ^2=103*8 mm. on drawing

= 4*32 mm. actual

(jy= 3*084 intervals = 18*5 mm. on drawings 0*771 mm. actual

To calculate the value of ^7} is first written in every compart-

ment of the table against the corresponding frequency, treating the class-
interval as unit. In Table 9.9 frequencies are shown in ordinary type
and the values of J?; in heavy type. In making these entries the sign
of the product may be neglected, but it must be remembered that this
sign will be positive in the upper left-hand and lower right-hand quadrants,
and negative in the two others. The frequencies are then collected,
according to the magnitude and sign of in columns 2 and 3 of Table
9.10. When columns 2 and 3 are completed they should be checked
to see that no frequency has been dropped, which may readily be done
by adding together the total of the two columns and the frequency
in the 8th row and 8th column of Table 9.9 (the row^ and column for
w^hich gi;=0), care being taken not to count twice the frequency in the
compartment common to the two. This grand total must clearly be
equal to N, the total number of observations, which in this case is 266.
The numbers in column 4 are given by deducting the entries in column 3
from those in column 2, The totals so obtained are multiplied by iif
(column 1) and the products entered in column 5 or 6 according to sign.
The algebraic sum of these totals gives

S(^^)=-+1519*5

228

THEORY OE STATISTICS

TABLE 9.10

2 3

Frequencies

Total

5 6

Products

Quadrants

8*5

~ 8*5

—

8*5

13*5

-f 3*5

—

10*5

-f 1*5

4*5

13*5

6*5

+ 7

—

0*5

+ 1*5

7-5

13-5

+ 8*5

—

+ 12

—

+ 5

6-5

+ 5*5

17*5

—

+ 17*5

210

—

+ 1

—

+ 6

—

+ 7

112

—

+ 2

—

+ 8

160

—

+ 2

—

+ 6

144

—

+ 1

—

+ 1

—

+ 3

—

+ 1

—

+ 1

—

+ 2

—

■+ 1

—

+ 1

—

Totals

145*5

71*5

266

+ 1,528
~ 8*5

1,519*5

- 8*5

Hence, dividing by 266,

ls(g7)=5-712

^=5-712-|;y=5-712-0-215

=5-497

Hence.

5-497

2-828 x 3-084

-0-63

The regression of daughter-frond on mother-frond is 0-69 (a value
which will not be affected by altering the units of measurement for both
mother- and daughter-fronds, as such an alteration will affect both
standard deviations equally). Hence, the regression equation giving the

CORRELATION AND REGRESSION 229

average actual length (in millimetres) of daughter-fronds for mother-fronds
of the actual length X is

Y=1.48+0*69Z

We leave it to the student to work out the second regression equation
giving the average length of mother-fronds for daughter-fronds of length 7,
and to check the whole work by a diagram shoving the lines of regression
and the means of arrays for the central portion of the table.

Example 9.3, Table 9.2. — The following device is frequently useful,
and saves a considerable amount of labour in calculating the product
term

We have —

. . . (i)

and

S(^+y)2=S(^2)+2S(xy)+-2(y2) . . . (ii)

Hence, knowing S(a;^) and we can find Ei[xy) if we know either

or These quantities are often easier to calculate than

itself.

Consider the data of Table 9.2. In the usual way, taking a working
mean centred in the intervals X =25- years, Y =25- years, we have, in
units of five years —

1=4-0*2924 ^ = -0*2353

i;(g2) =9,708 2(7?^) =7,090

a;.=l*730 ay=*481

Now the value of ^ is constant doivn diagonals which run from the
top left hand to the bottom right hand of the table. In fact, for the
principal diagonal, running from Z=15-,Y=15- through Z=20-,
Y=20-, etc., ^—^=0. For the diagonal above this, running from
X=20~, Y=15- through .ST =25-, Y=20-, etc., g— ^=1, and so on.
Let us then find the diagonal totals. We find —

g-7

Frequency in
diagonal

-3

-2

-1

280

1,398

1,051

263

3,153

230

THEORY OF STATISTICS

The total is the total frequency, which gives a check on the work.

The value of for the whole table is then obtained from the

above table by squaring the values in the left-hand column, multipl 3 dng
by the corresponding frequency in the right-hand column and addmg.
We get

X(g-.^)2=:(9x4)+(4x34) + (lx280)+ . . . +{49 x2)

=4,286

Hence, from (i),

4,286=9, 708+7,090-2S(g^)

/. =6,256

+ 2-0529

whence

2-0529

■*■1 -730x1 *481

0-80

The regression equations may now be obtained in the usual manner.

In the above work we chose equation (i) in preference to equation (ii)
because the frequencies are seen by inspection to run mainly from the
top left hand to the bottom right hand of the table. Had they run from
the top right hand to the bottom left hand we should probably have found
it better to use equation (ii).

9.27 The student should be careful to remember the following points
in working —

(1) To give and their correct signs in finding the true mean
deviation product p,

(2) To express Qx and Oy in terms of the class-interval as a unit, in the
value of r—pjux^y, for these are the units in terms of which p has been
calculated.

(3) To use the proper units for the standard deviations (not class-
intervals in general) in calculating the coefficients of regression : in forming
the regression equation in terms of the absolute values of the variables,
for example, as above, the work will be wrong unless means and standard
deviations are expressed in the same units.

Fluctuations of sampling

9.28 Further, it must always be remembered that correlation coefficients,
like other statistical measures, are subject to fluctuations of sampling.
We shall consider this point at some length in later chapters (18 and 21),
since the correlation coefficient has certain individual features which
make it of special interest from the sampling point of view. We may,
however, at this stage stress that if the number of observations is small,
no significance can be attached to small, or even moderately large, values
of f as indicating a real correlation in the population from which the

CORRELATION AND REGRESSION

231

observations are drawn. For example, if N =36, a value of r 5 may
be a chance result, though a very infrequent one, m sampling from an
uncon-elated population. If iV=100, r=±0-S may similarly be a mere
fluctuation of sampling, though again a very infrequent one. The student
should therefore be careful in interpreting his coefficients.

Corrections for grouping

9,29 In this connection we may mention the question whether, in calcu-
lating the correlation coefficient from grouped data, any correction is
to be made analogous to the Sheppard correction for grouping which
we have considered in the case of univariate data. In the examples
considered in the foregoing we have not made such corrections.

It appears that, when the distribution is reasonably symmetrical and
obeys conditions similar to those enunciated in 6.12, page 133, we may,
with advantage, correct the standard deviations (Jx, cry, by applying to
each the formula

O ’ 2 (corrected) =o-2— --

X M

where h is the width of the interval. The product term ^{xy) needs no
such correction.

We pointed out in 6.12, however, that sampling fluctuations usually
obliterate any correction for grouping unless the size of the sample is large.
It may, as before, be suggested that unless 1,000 or more, it is hardly
worth while making the correction. For example, in Tables 9. 1-9.6,
Tables 9.1 and 9.5 have a frequency less than 1,000 and the corrections
are not to be applied — in any case they would not be applied to Tables
9.5 and 9.6, which violate the conditions as to tapering off/'

9;30 FinaDy, it should be borne in mind that any coefficient, e.g. the
coefficient of correlation or the coefficient of contingency, gives only a
part of the information afforded by the original data or the correlation
table. The correlation table itself, or the original data if no correlation
table has been compiled, should always be given, unless considerations of
space or of expense absolutely preclude the adoption of such a course.

SUMMARY

1 . A population every member of which bears one of the values of each
of two variates is said to be bivariate. If the members are grouped
according to class-intervals of the two variables, we have a bivariate
frequency-distribution.

2. The bivariate frequency-distribution may be represented by a
frequency-surface or by a stereogram. Ungrouped data (and, less con-
veniently, grouped data) can be represented on a scatter diagram.

232

THEORY OF STATISTICS

3. The means of arrays of a bivariate frequency-distribution may be
represented as points by reference to a pair of rectangular axes along
which are measured values of the variables. The means of rows and
those of columns will in general lie respectively about two smooth curves,
called lines of regression. The equations of these curves are called
regression equations.^

4. The regression equations may be regarded as expressions for
estimating from a given value of one variate the average corresponding
value of the other.

5. The coefficient of correlation (product-moment correlation coefficient)
between two variables X and Y is given by —

J:(xy}

^ VS(x^)i:(y^)

_ P

CTjfOy

where x, y are the values of the variables measured from their respective
means, and

6. The correlation coefficient r cannot be less than —1 or greater than

+1. If r=±l the variables are perfectly correlated, the points corre-
sponding to pairs of values x, y all lying on a straight line. If 1
the variables are perfectly negatively correlated, low values of one
corresponding to high values of the other. If the variables aie

perfectly positively correlated, high values of one corresponding to high
values of the other.

7. The linear regression equation of X on F (referred to axes through
their respective means) is

where

and that of Y on is
where

and 6^ being called coefficients of regression, or simply regressions.

x=b^y
h — P

(Ty ~Ofy®
y—bfX

A P

H 7^2

(Tx

^ Curviimear regression lines, like straight regression lines, may also be defined for
imgrouped data by an extension of the pnnciple of making sums of squares of errors of
estimate a minimum.

CORRELATION AND REGRESSION

233

8 . The straight lines of regression are such that the sums of squares
of errors of estimate, and 62 ^) 2 , are a minimum. If the

quotients of these sums by N are denoted by

EXERCISES

Find the correlation coefficient and the equations of regression for the
following values of X and Y —

X Y

1 2

2 5

3 3

4 8

5 7

[As a matter of practice it is never worth calculating a correlation coef-
ficient for so few observations : the figures are given solely as a short
example on which the student can test his knowledge of the work.]

9.2 (Data from W, Little : Labour Commission Report, Vol. 5, Part 1,
1894, and Official Returns.)

The figures in the table on p. 234 show (1) the estimated average eamings
of agricultural labourers, X, (2) the percentage of population in receipt of
poor law relief, Y, (3) the ratio of the number of paupers receiying outdoor
relief to the number receiving relief in workhouses, Z, for certain districts
in England and Wales in 1^93.

Find the correlations between X and Y, Y and Z, and Z and X. Draw
scatter diagranas to illustrate the various joint distributions. '

9.3 Verify the data in the table heading p. 235 for the under-mentioned
tables of this chapter. Calculate the means of rows and columns and
draw a diagram showing the lines of regression for the data of Table 9.1
(Sheppard’s correction used only in Table 9.4.)

In calculating the coefficient of contingency (coefficient of mean square
contingency) use the following groupings, so as to avoid small scattered
frequencies at the extremities of the tables and also. excessive arithmetic —

Table 9.L Group together (1) two top rows, (2) three bottom rows,
(3) two first columns, (4) four last columns, leaving centre of table as
it stands.

THEORY OF STATISTICS

Table for Exercise 9.2

Union

Estimated
average earnings
of agricultural
labourers
Shillings and
pence per week

Percentage of
population in
receipt of
Poor Law
relief

Ratio of number
of paupers
receiving
outdoor relief
to the number
receiving relief
m workhouses

1. Glendale .

2*40

6*40

2. Wigton

2-29

4*04

3. Garstang .

1*39

7*90

4. Helper

1-92

3*31

5. Nantwich .

2-98

7*85

6. Atcham

1-17

0*45

7. Dnffield .

3-79

10*00

8. Uttoxeter .

3*01

4*43

9. Wetherby

2-39

4*78

10. Easmgwold

2-78

4*73

11. Southwell

3-09

6*66

12. Hollmgbourn

2-78

1*22

13. Melton Mowbray

2*61

4*27

14. Truro

4-33

7*50

15. Godstone .

3*02

4*44

16. Louth

4*20

8*34

17. Bnx worth

1-29

0*69

18. Crediton .

5*16

9*89

19. Holbeach .

4*75

4*00

20. Maldon

4*64

6*02

21. Monmouth,

4*26

8*27

22. St. Neots .

1*66

! 1*58

23. Swaffham .

5*37

i 16*04

24. Thakeham

3*38

1*96

25. Thame

5*84

9*28

26. Thmgoe

4*63

8*72

27. Basingstoke

3*93

2*97

28. Cirencester

4*54

5*38

29. North Witchford

3*42

3*24

30. Pewsey

5*88

7*61

31. Bromyard

4*36

5*87

32. Wantage .

3*85

5*50

33. Stratford-on-Avon

3*92

3*58

34. Dorchester.

4*48

6*93

35. Woburn

5*67

6-02

36. Buntmgford

4*91

4*92

37. Pershore .

4*34

4-64

38. Langport .

5*19

10*56

Table 9.3. Regroup by 2-inch intervals, 58 ‘5-60 *5, etc., for father,
59*5-61 -5, etc., for son. If a 3-inch grouping be used (58*5-61*5, etc.,
for both father and son), the coefficient of mean square contingency is
0-465.

Table 9.4. For columns, group those headed 3 and 4, 5 and 6, 7 and 8,
9 and 10, 11 and over ; for rows, group those headed 8-11, 12-13, 14-15,
16-17,18-19, 20-21, 22-23, 24-25, 26-27, 28 and over.

CORRELATION AND REGRESSION

235

Table for Ezerdse 9.3

9.1

9.6

Mean of X .

Standard deviation of X

„ „ y .

Coefficient of correlation

Coefficient of contingency'!
(for the grouping ^ted >
below) j

55*3 mm.
53- 1 „

6*86 „
5-77 „
4-0-97

67*70 in
68*66 „
2*72 „
2-75 „
4-0-51

6*22 yrs
18*61 gal
2*21 yrs
3*37 gal
+0*22

14*54 per thou
379*47 births

2 • 87 per thou
505 * 24 births
+0*17

0*90

0 51

0*26

0*30

Table 11.6. For columns, take singly those for 0~, 200~, group 400-
and 600- and group 800- and over. Rows, group those headed 6-11,
12 and 13, 14 and 15, 16-18, 19 and over.

9.4 (Data from Statistical Review of England and Wales for 1933, Tables,
Part 1, p. 3, and part 2, p. 6.) The following show mean annual birth
and death rates in England and Whales for quinquennia since 1876. Find
the correlation between birth and death rates.

Period

Mean annual

Live birth rate
per 1,000 of population

Mean armuaJ
death rate

per 1,000 of population

1876-80

35*3

20*8

1881-85

33*5

19*4

1886-90

31*4

18*9

1891-95

30*5

18*7

1896-1900 1

29*3

17*7

1901-1905

28*2

16*0

1906-1910

26*3

14*7

1911-15

23*6

14*3

1916-20

20*1

14*4

1921-25

19*9

12*2

1926-30

16*7

12*1

9.5 The following figures (S. Rowson, Journ. Roy. Sta. Soc., vol. 99, 1936),
give the relationship between the density of population and seating capacity
of cinemas in various districts of Great Eritain.

Find the correlation between density of population and proportion of
cinemas with (1) seating capacity 500 or less, (2) seating capacity 2,000
or more.

236

THEORY OF STATISTICS

District

Density of
population
per square mile

Percentage of cmemas

(1)

Seating 500
or less

(2)

Seating 2,000
or more

Scotland

163

13-4

4*3

North Wales ....

165

42-5

0-0

West of England .

380

38-2

2-1

Eastern Counties .

431

38*8

1*3

South Wales ....

440

22-4

1*2

North of England

487

16-0

1*2

Yorkshire and district .

594

15*5

3-1

Midlands

710

20-2

1-6

Home Counties (excl London)

794

28 2

3-0

Lancashire

2.157

13-5

i 3*6

9.6 Show that the coefficient of correlation is the geometric mean of the
coefficients of regression; verify from the data of Examples 9.1, *9.2
and 9.3 that the arithmetic mean of the coefficients of regression is
greater than the coefficient of correlation.

9.7 The tangent of the difference of angles A and B is given hy —

tan {A—B)

tan .4— 'tan B
1 +tan A tan B

Deduce that the smaller angle between regression lines is 0, given by —

tan 6'

1 — /2

and interpret this result when r=0 and r=±l«

CHAPTER TEN

NORMAL CORRELATION

The bivariate normal surface

10.1 Our study of the normal curve in Chapter 8 may be extended
to yield a corresponding expression for the frequency-distribution of pairs
of values of two variates. This bivariate normal distribution, known also
as “ the bivariate normal surface” “ the normal correlation surface or
simply “ the normal surface” occupies a central position in the theory
of bivariate frequency-distributions, and bears to them a relation similar
to that borne by the normal curve to the frequency-distributions of a
single variate.

The norfnal surface is of great historical importance, as the earlier
work on correlation is, almost without exception, based on the assumption
of such a distribution ; though when it was recognised that the properties
of the correlation coefi&cient could be deduced, as in Chapter 9, without
reference to the form of the distribution of frequency, a knowledge of this
special type of frequency-surface ceased to be so essential. But the
generalised normal law is of importance in the theory of sampling : it
serves to describe very approximately certain actual distributions (e.g. of
measurements on man) ; and if it can be assumed to hold good, some of the
expressions in the theory or correlation, notably the standard deviations
of arrays (and, if more than two variables are involved, the partial correla-
tion coefficients), can be assigned more simple and definite meanings than
in the general case. The student should, therefore, be familiar with the
more fundamental properties of the distribution.

10.2 Consider first the case in which the two variables are completely
independent. Let the distributions of frequency for the two variables
Xi and singly be given by

yi=3'i'exp(-%V2ai*)|

3'a=3'2'exp(-*a^'2o•^^j

Then, assuming independence, the frequency-distributions of pairs of
values must, by the rule of independence, be given by

*37

( 10 . 2 )

Equation (10.2) gives a normal correlation surface for one special case, the
correlation coefl&cient being zero. If we put constant, we see that
every section of the surface by a vertical plane parallel to the ^r^-axis, i.e.
the distribution of any array of %'s, is a normal distribution, with the same
mean and standard deviation as the total distribution of %’s ; and a similar
statement holds for the arrays of ; these properties must hold good,
of course, as the two variables are assumed independent (cf. 3,18). The
contour lines of the surface, that is to say, lines drawn on the surface at a
constant height, are a series of similar ellipses with major and minor axes
parallel to the axes of % and x^ and proportional to % and Og, the equations
to the contour lines being of the general form

(10.4)

Pairs of values of % and ^2 related by an equation of this form are, therefore,
equally frequent.

10.3 Now suppose we have two correlated variates % and and let
the regression of % on X 2 be ^hat of X 2 on % be Eet be the

coefficient of correlation between % and

Consider the new variates defined by the equations

%. 2 =%— ^ 12^2
X2i=X2 ^ 21^1

This is a notation which we shall later extend considerably.

Then % and x^.^ are uncorrelated, as are ^2 2 * ’

For

S (AJjATg i) =2 -^Xi {x2—b 21 %) }

-S(%^2^--62iS(%)^

and similarly for

Writing %, for the standard deviations of %, we see that the
Standard deviation ^1.2 ^ given by

NORMAL CORRELATION

239

={(7f-26ijri2aia2+6J2a|}

={(yl-2rl,(jl+rli<Tl}

and similarly the standard deviation of % 2 .i is given by

a|.i=a|(l-rf2)

We. obtained these results in a slightly different form in 9.22 and 9.24.

10.4 Suppose further that and are not only uncorrelated, but
independent, and that each is normally distributed.

In accordance with equation (10.2), we must have for the frequency-
distribution of pairs of deviations of and ^

But

>'i2=3''i 2 exp

. (10.5)

I ^2 1 -

aj ct| I

' <yUl-r

-2r.

X-iXq

12,

0-102(1-

'121

-2^18

Evidently we should also have arrived at precisely the same expression
if we had taken the distribution of frequency for and x^ ^^ and reduced
the exponent

?4.
2 »

^1.2

iw 2

^1.2

We have, therefore, the general expression for the normal correlation
surface for two variables —

. . ( 10 . 6 )

I \*^1.2 ^ 2.1 <^1.2°^2.l/ j

Further, since Xi and ^g.i. ^2 %.2. independent, we must have :

N N N

yi2--

'2710102.1 2770201.2 2^0-102(1 — rfg)*

. (10.7)

Expressing o-j 2 <^ 2.1 ^ terms of a^, and we have the

alternative form

yi2=

27rai02 V 1

'12

I 1 /% 27-i2a:i:ra , a;|\]

240

THEORY OF STATISTICS

Properties \)f the normal surface

10,5 For any given value of the distribution of the array of
is given by

This is a normal distribution of standard deviation g, with a mean
deviating by from the mean of the whole distnbution of

Hence, since may be any value, we have the important results —

Fig. 10.1. — ^Principal axes and contour lines of the normal
correlation surface

NORMAL CORRELATION 24I

(1) that the standard deviations of all arrays of % are the same, and
equal to 0*3^. g ;

(2) that the regression of on x^ is strictly linear.

Similarly, it follows that the s.d/s of all arrays of X 2 are equal to cX 2 . 1 t
and that the regression of x^ on x^ is linear.

10.6 The contour lines are, as in the case of independence, a series
of concentric and similar ellipses ; the major and minor axes ar^, however,
no longer parallel to the axes of x^ and x^, but make a certain angle with
them. Fig. 10.1 illustrates the calculated form, of the contour lines for
one case, RR and CC being the lines of regression. As each line of re-
gression cuts every array of x^ or of X 2 in its mean, and as the distribution
of every array is symmetrical about its mean, RR must bisect every
horizontal chord and CC every vertical chord, as illustrated by the two
chords shown by dotted lines ; it also follows that RR cuts all the ellipses
in the points of contact of the horizontal tangents to the ellipses, and CC in
the points of contact of the vertical tangents. The surface or solid itself,
somewhat truncated, is shown in fig. 9.1, page 208.

10.7 Since, as we see from fig. 10.1, a normal surface for two correlated
variables may be regarded merely as a certain surface for which r is zero
turned round through some angle, and since for every angle through which
it is turned the distributions of all % arrays and X 2 arrays are normal, it
follows that every section of a normal surface by a vertical plane is a normal
curve, i.e. the distributions of arrays taken at any angle across the surface
are normal.

10.8 It also follows that, since the total distributions of and X 2 must
be normal for every angle through which the surface is turned, the
distributions of totals given by slices or arrays taken at any angle across a
normal surface must be normal distributions. But these would give the
distributions of functions like ± bx 2 , and consequently (1) the dis-
tribution of any linear function of two normally distributed variables Xi
and X 2 must also be normal ; (2) the correlation between any two linear
functions of two normally distributed variables must be normal correlation.

Result (1) is very important, and may easily be extended to
caver the case of n variables x^ , , , Xn. Suppose, in fact, we have
n such variables each of which is normally distributed, and a linear
function ax^-\-hx 2 + . . . +hxn. Since ax^-\-bx 2 is normally distributed,
(ajTi+teg) +^^3 is normally distributed, and hence so is {ax^bx 2 + 0 X 2 )
and so on. Thus the function is normally distributed.

Hence, the sum of n normal variates is distributed normally ; and in
particular the mean of n normal variates is distributed normally. More
particularly still, the means of samples of n from a normal population are
normally distributed.

242

THEORY OF STATISTICS

10.9 Returning to the normal surface, it is interesting to inquire what

IS the angle d through which the surface has been turned from the position
for which the correlation was zero. The major and minor axes of the
ellipses are sometimes termed the pnnctpal axes. If ^2 co-

ordinates referred to the principal axes (the ^^-axis being the ^r^-axis in
its new position), we have for the relation between the angle

0 being taken as positive for a rotation of the ;i;i-axis which will make it,
if continued through 90®, coincide in direction and sense with the A; 2 “axis,

cos Q +^2 sin ^ ^ qq gv

cos d—x^ sin ’ ‘ * V • ;

But, since ^ 1 , ^2 sire uncorrelated, Hence, multiplying together

equations (10.9) and summing,

0 = (cr 2 ^— o-^^) sin 26 -\- 2 r^<p-^G 2 , cos 26

tan 2(9=— ^ - " -^-^ .... (10.10)

It should be noticed that if we define the principal axes of any distribution
for two variables as being a pair of axes at right angles for which the
variables gg are uncorrelated, equation (10.10) gives the angle that they
make with the axes of measurement whether the distribution be norm^
or not.

10.10 The two standard deviations, say and Sg, about the principal
axes are of some interest, for evidently from 10.2 the major and minor
axes of the contour ellipses are proportional to these two standard
deviations. They may be most readily determined as follows. Squaring
the two transformation equations (10.9), summing and adding, we have —

S,2+S22=ai2+o-22 .... (10.11)

Referring the surface to the axes of measurement, we have for the central
ordinate, by equation (10.7),

, N

^ 2na^(y^{\—rl^^

Referring it to the principal axes, by equation (10.3),

2775,58

But these two values of the central ordinate must be equal, therefore

S,S^==a,cr,(l^ri,)i . . . (10.12)

(10.11) and (10.12) are a pair of simultaneous equations from which and
Sg may be very simply obtained in any arithmetical case. Care must,
however, be taken to give the correct signs to the square root in solving.

NORMAL CORRELATION

243

S 1 +S 2 is necessarily positive, and also if r is positive, the major

axes of the ellipses lying along ; but if r be negative, S^—S^ is also
negative. It should be noted that, while we have deduced (10.12) from
a simple consideration depending on the normality of the distribution, it
is really of general application (like equation (10.1 1)), and may be obtained
at somewhat greater length from the equations for transforming co-
ordinates.

10.11 As an example of the application of the foregoing theory to a
practical case, we proceed to consider the distribution of Table 9.3,
page 202, showing the correlation betw^een stature of father and son, and
to test, as far as we can by elementary methods, whether a normal surface
will fit the data.

10.12 The first important property of the normal distribution is the
linearity of regression. This was well illustrated for these data in fig. 9.8
(page 218). Subject to some investigation as to the deviations from strict
linearity which may occur as the result of sampling fluctuations, we may
conclude that the regression is appreciably linear. We shall consider a
test of linearity in later chapters (see Chapter 21).

10.13 The second important property is the constancy of the standard
deviation for all parallel arrays.

The standard deviations of the ten columns from that headed 62 *5-63 *5
onwards are —

2-56

2-60

2-11

2-26

2-55

2-26

2-24

2-45

2-23

2-33

the mean being 2*36. The standard deviations agam only fluctuate
irregularly round their mean value. The mean of the first five is 2-34, of
the second five 2-38, a difference of only 0*04 ; of the first group, two are
greater and three are less than the mean, and the same is true of the second
group. There does not seem to be any indication of a general tendency
for the standard deviation to increase or decrease as we pass from one end
of the table to the other. We are not yet in a position to test how far the
differences from the average standard deviation might have arisen in
sampling from a record in which the distribution was strictly normal, but,
as a fact, a rough test suggests that they might have done so.

10.14 Next we note that the distributions of all arrays of a normal
surface should themselves be normal. Owing, however, to the small
numbers of observations in any array, the distributions of arrays are very
irregular, and their normality cannot be tested m any very satisfactory
way ; we can only say that they do not exhibit any marked or regular
asymmetry. But we can test the allied property of a normal correlation

244

THEORY OF STATISTICS

table, viz, that the totals of arrays must give a normal distribution even
if the arrays be taken diagonally across the surface, and not paraUel to
either axis of measurement. From an ordinary correlation table we
cannot find the totals of such diagonal arrays exactly, but the totals of
arrays at an angle of 45° will be given with sufficient accuracy for our
present purpose by the totals of lines of diagonally adjacent compartments.
Referring again to Table 9.3, and forming the totals of such diagonals
(running up from left to right), we find, starting at the top left-hand corner
of the table, the following distribution —

0-25

78-75

81-25

3-25

66-5

6-25

59-25

42-25

9-75

30-75

29-25

34-5

10-75

46-25

60-5

4-25

’67-5

3-5

85-75

1-75

87-25

0-25

94-25

Total

1078

The mean of this distribution is at 0*359 of an interval above the centre of
the interval with frequency 78 ; its standard d!s\dation is 4 • 757 intervals, or,
remembering that the interval is 1 jV2 of an inch, 3*364 inches. (This
value may be checked directly from the constants for the table given in
Exeicise 9.3, page 235, for we have, from the first of the transformation
equations (12.9),

cos^ sin^ sin d cos 6

and inserting o-i==2*72, <72=2*75, rj2=0*51, sin ^=cos 0=1 /V2, find
0^=3 *361.) Drawing a diagram and fitting a normal curve, we have
fig. 10.2 ; the distribution is rather irregular but the fit is fair ; certainly
there is no marked asymmetry, and, so far as the graphical test goes, the
distribution may be regarded as appreciably normal. One of the greatest
divergences of the actual distribution from the normal curve occurs in the
almost central interval with frequency 78 ; the difierence between the
observed and calculated frequencies is here 12 units, but nevertheless it

normal correlation

^45

Fig. 10,2. — Distribution of frequency obtained by addition of Table 9.3 along diagonals
running up from left to right, fitted vsdth a normal curve

may well have occurred as a fluctuation of sampling. In fact, anticipating
our discussion of the use of the standard error (standard deviation of
simple' sampling) in testing the significance of sampling fluctuations
(17,4), we may note that the standard error in this case is Vnpq, where
n IS the number of observations and p and q the chances of an individual
falling or not falling within the given interval, p may be taken as 90 /1 078,
and therefore the standard error is

90 988

The observed deviation, 12, is not much greater than this and may there-
fore have occurred as a sampling fluctuation. We have used here the
exact expression for the standard error, but since p is small we might
have used the approximation “V pn—V%0^d-S, This last is useful as
giving a test which can be applied on sight

10.15 So far, we have seen (1) that the regression is approximately
linear ; (2) that, in the arrays which we have tested, the standard
deviations are approximately constant, or at least that their differences
are only small, irregular and fluctuating ; (3) that the distribution of
totals for one set of diagonal arrays is approximately normal. These
results suggest, though they cannot completely prove, that the whole
distribution of frequency may be regarded as approximately normal,
within the limits of fluctuations of sampling. We may therefore apply a
more searching test, viz the form of the contour lines and the closeness
of their fit to the contour ellipses of the normal surface. It may, however,
be seen that no very close fit can be expected. Since the frequencies in
the compartments of the table are small, the standard error of any
frequency is given approximately by its square root (17.15), and this

246

THEORY OF STATISTICS

implies a standard error of about 5 units at the centre of the table, 3 units
for a frequency of 9, or 2 units for a frequency of 4 ; fluctuations of these
magnitudes are quite possible and might cause wide divergences in the
corresponding contour lines.

10.16 Using the suffix 1 to denote the constants relating to the distribu-
tion of stature for fathers, and 2 the same constants for the sons,

A7=1078 Mi=67-70 71^2=68-66

cri=2-72 CT2=2-75

Hence we have from equation (10.7),

y'i3=26-7

and the complete expression for the fitted normal surface is

exp

l 'V

The equation to' any contour ellipse will be given by equating the index
of 5 to a constant, but it is very much easier to draw the ellipses if we refer
them to their principal axes. To do this we must first determine d,
and Sg. From (10.10),

tan 20= -46-49

whence 20=91® 14', 0=45® 37', the principal axes standing very nearly
at an angle of 45® with the axes of measurement, owing to the two standard
deviations being very nearly equal. They should be set off on the diagram,
not with a protractor, but by taking tan 0 from the tables (1.022) and
calculating points on each axis on either side of the mean.

To obtain 5^ and Sg we have, from (10.11) and (10 12),

5 ^ 2 + 522 = 14-961

25iS2==U868

Adding and subtracting these equations from each other and taking the
square root,

Si+S2=5-275

-52=1-447

whence 5i=3-36, 52=1-91 ; owing to the principal axes standing nearly
at 45® the first value is sensibly the same as that found for og in 10,14.
The equations to the contour ellipses, referred to the principal axes, may
therefore be written in the form —

(3-36)*^ (I -91)*

NORMAL CORRELATION

247

the major and minor semi-axes being 3* 36 X r and 1 * 91 X c respectively. To
find c for any assigned value of the frequency y we have —

, 2(logy'i2-log:yta)

log e

Supposing that we desire to draw the three contour ellipses for y=5,
10 and 20, we find c=l -83, 1 *40 and 0-76, or the following values for the
major and minor axes of the ellipses : semi-major axes, 6- 15, 4*70, 2* 55 ;
semi-minor axes, 3-50, 2-67, 1*45. The ellipses drawn with these axes
are shown in fig. 10.3, very much reduced, of course, from the original

Fig. 10.3, — Contour for the frequencies 5, 10 and 20 of the distribution of Table

11,3, and corresponding contour ellipses of the fitted normal surface
PiPj, PjPg, principal axes ; M, mean.

drawing, one of the squares shown representing a square inch on the
original. The actual contour lines for the same frequencies are shown
by the irregular polygons superposed on the ellipses, the points on these

248 THEORY OF STATISTICS

polygons having been obtained by simple graphical interpolation between
the frequencies in each row and each column — diagonal interpolation
between the frequencies in a row and the frequencies in a column not
being used. It will be seen that the fit of the two lower contours, is on
the whole, fair, especially considering the high standard errors. In the
case of the central contour, y =20, the fit looks very poor to the eye, but
if the ellipse be compared carefully with the table, the figures suggest
that here agam we have only to deal with the effects of fluctuations of
sampling. For father's stature =66 in., son's stature =70 in., there is a
frequency of 18*75, and an increase in this much less than the standard
error would bring the actual contour outside the ellipse. Again, for
father’s stature=68 in. son's statute=71 in., there is a frequency of 19,
and an increase of a single unit would give a point on the actual contour
below the ellipse. Taking the results as a whole, the fit must be considered
quite as good as we could expect with such small frequencies.

Isotropic character of the normal surface

10.17 The normal distribution of frequency for two variables is an
isotropic distribution, to which aU the theorems of 3.16 apply. For
if we isolate the four compartments of the correlation table common
to the rows and columns centring round values of the variables X 2 ,
^ 1 , ^ 2 ^ we have for the ratio of the cross-products (frequency of x^x^
multiplied by frequency of Xj,'x 2 ', divided by frequency of X 1 X 2 ' multiplied
by frequency of

(>xp;:r ~ —(Xi'-Xi)(xs' -x^)

^ 1 . 2 ^ 2.1

Assuming that has been taken of the same sign as

exponent is of the same sign as Hence, the association for this group
of four frequencies is also of the same sign as ^^itio of the cross-

products being unity, or the association zero, if is zero. In a normal
distribution, the association is therefore of the same sign — ^the sign of
^ 12 — every tetrad of frequencies in the compartments common to
two rows and two columns ; that is to say, the distribution is isotropic.
It follows that every grouping of a normal distribution is isotropic whether
the class-intervals are equal or unequal, large or small and the sign of the
association for a normal distribution grouped down to 2 x 2-fold form
must always be the same whatever the axes of division chosen,

10.18 These theorems are of importance in the applications of the
theory of normal correlation to the treatment of qualitative characters
which are subjected to a manifold classification. The contingency tables
for such characters are sometimes regarded as groupings of a normal
distribution of frequency, and the coefi6cient of correlation is determined
on this hypothesis by a special procedure (see below, 11.29, page 268).

NORMAL CORRELATION

249

Before appl 5 dng this procedure it is well, therefore, to see whether the
distribution of frequency may be regarded as approximately isotropic,
or reducible to isotropic form by some alteration in the order of rows
and columns (3.16 and 3.17). If only reducible to isotropic form by
some rearrangement, this rearrangement should be eifected before grouping
the table to 2 x 2-fold form for the calculation of the correlation coefficient
by the process referred to. If the table is not reducible to isotropic
form by any rearrangement, . the process of calculating the coefficient
of correlation on the assumption of normality is to be avoided. Clearly,
even if the table be isotropic it need not be normal, but at least the test
for isotropy affords a rapid and simple means for excluding certain dis-
tributions which are not even remotely normal. Table 3,2, page 50,
might possibly be regarded as a grouping of.normally distributed frequency
if rearranged as suggested in 3.15 — it would be worth the investigator's
while to proceed further and compare the actual distribution with, a fitted
normal ffistribution — but Table 3.4 could not be regarded as normal, and
could not be rearranged so as to give a grouping of normally distributed
frequency.

10.19 If the frequencies in a contingency table be not large, and also
if the contingency or correlation be small, the influence of casual irregu-
larities due to fliuctuations of sampling may render it difficult to say
j whether the distribution may be regarded as essentially isotropic or
not. In such cases some further condensation of the table by grouping
together adjacent rows and columns, of some process of “ smoothing”
by averaging the frequencies in adjacent compartments, may be of service.
The correlation table for stature in father and son (Table 9*3), for instance,
is obviously not strictly isotropic as it stands : we have seen, however,
that it appears to be normal, vdthin the limits of fluctuations of sampling,
and it should consequently be isotropic within such limits. We can
apply a rough test by regrouping the table in a much coarser form, say
with four rows and four columns : the table below exhibits such a grouping,

TABLE 10.1 — (Condeiiised from Table 9.3, p. 202)

Son's stature
{inches)

Under 66*5
66-5-68-5
68-5-70-5
70-5 and over

Father's stature (inches)

65 . 5 ^ 7.5 67-5-69.S JlXer

Total

97-5 74-25 34-75 10-5

76-5 108 85 52

33-25 64-75 95 84-5

14-75 32-5 80-75 134

217

321-5

277-5

262

Total

222 279-5 295-5 281

1,078

250

THEORY OF STATISTICS

the limits of rows and of columns having been so fixed as to include not
less than 200 observations in each array.

Taking the ratio of the frequency in column 1 to the sum of the frequencies
in columns 1 and 2 for each successive row, and so on for the other pairs of
columns, we find the following series of ratios —

TABLE 10.2 — Ratio of frequency in column m to frequency in column m plus frequency
in column (m+l) of Table 10.1

Row

1 and 2

Columns

2 and 3

3 and 4

0-568

0-681

0-768

0-415

0-560

0-620

0-339

0-405

0-529

0*312

0-287

0*376

These ratios decrease continuously as we pass from the top to the bottom
of the table, and the distribution, as condensed, is therefore isotropic.
The student should form one or two other condensations of the original
table to 3- x3- or 4- x 4-fold form : he will probably find them either
isotropic or diverging so slightly from isotropy that an alteration of the
frequencies, well within the margin of possible fluctuations of sampling,
will lender the distribution isotropic.

Relationship between contingency and normal correlation
10.20 It was shown by Karl Pearson that if a normal bivariate population
is divided into sections so as to form a contingency table, the coefficient
of mean square contingency, C, tends to the value r in magnitude as the
intervals become finer and finer, though of course it is always positive
in sign It was, in fact, the relation

where is the mean-square contingency, which led Pearson to identify
C with the expression on the right.

The values of C and t for the distributions of some of the tables of
Chapter 9 were compared in Exercise 9.3, page 235.

NORMAL CORRELATION

251

SUMMARY

1. The equation of the normal surface is

1 /xf x\\\

2(1 o-iO-a ajVJ

where is the s.d. of that of x^, and the correlation between

and x^.

This may also be written

„ N^/l—r^ [ J xl 2ri2XiXz , xl \

yi2=7r^ exp • -i -r

27rcri.2CT2i Vfff,2 <^i.2'^2.i <^ 1 , 1 /

where

:vi2=

27r<Jiao'\/i ■

-rh

exp

-''12). C 7 |.i=-a|(l -rfa)

2. For two variates normally correlated the standard deviations of
parallel arrays are equal and the regressions are linear.

3. Any section of the normal surface by a vertical plane is a normal
curve, and a section by a horizontal plane is an ellipse. The ellipses given
by horizontal sections are similar and similarly situated.

4. The bivariate normal distribution is isotropic.

5. A linear function of variates, each of which is normally distributed,
is also normally distnbuted.

EXERCISES

10.1 Deduce equation (10.12) from the equations for transformation of
co-ordinates without assuming the normal distribution.

10.2 Hence show that if the pairs of observed values of x^ and a ; 2 are
represented by points on a plane, and a straight line drawn through the
mean, the sum of the squares of the distances of the points from this line
is a minimum if the line is the major principal axis.

10.3 The coefficient of correlation with reference to the principal axes
being zero, and with reference to other axes something, there must be
some pair of axes at right angles for which the correlation is a maximum,
i.e. is numerically greatest without regard to sign. Show that these axes
make an angle of 45° with the principal axes, and that the maximum
value of the correlation is

, V-S2^

252

THEORY OF STATISTICS

10.4 (Sheppard, PUL Trans. Roy. Soc. A, 1898, 192, 101.) A fourfold
table is formed from a normal correlation table, taking the points of
division between A and a, B and fi, at the medians, so that {A) — (a) =(S)
=(/?) =:JV /2. Show that

/ 2{AB)\

10.5 Show that the points of inflection of the sections of the normal
surface by vertical planes through the mean of the distribution lie on an
ellipse ; and show how this ellipse may be used to give the standard devia-
tions of such sections.

10.6 Hence find the minimum and maximum standard deviations which
can be taken by such sections, and show that any specified value of the
s.d. between the minimum and maximum will be given by two, and only
two, sections.

10.7 Assuming that the heights of fathers and sons are distributed
in the bivariate normal form with a correlation which is positive but not
unity and with the same means and variances, show that fathers of more
than average height tend to have sons whose height, though above average,
is less than that of their respective fathers. Show also that sons of more
than average height tend to have fathers whose height is less than that
of their respective sons. Explain why these two results are not in-
consistent.

10.8 Find the conditions that the surface

z^k exp {ax^ -{-2hxy

can represent a normal correlation surface whose variates are x and y.
Assuming these conditions satisfied, express and iu terms of

a, k and b.

10.9 Corresponding to ;c-values, —n, —(n—l) . . . —1, 0,,1, . . . («— 1), n,
the y-values are the cubes of the A;-values. Show that the covariance
(9.25) of X and y is given by

/^n="r‘ + lower powers of n.

Hence show that for large n the correlation is approximately y'0*84=:
0*916 and thus is not unity although the variates are function^y related.

10.10 In a bivariate normal population the standard deviation of any
j!f-array is k times that of the ^-variate as a whole. Show that the correla-
tion is ^(1 —

CHAPTER ELEVEN

FURTHER THEORY OF CORRELATION

Methods of estimating the product-moment correlation coefficient

11.1 The only strict method of calculating the correlation coefficient
is that described in Chapter 10, from the formula

VS(;r2)2(y2)

Where possible this formula should be employed. It sometimes happens,
however, owing to incomplete data, that we are constrained to use some
method of approximation. Furthermore, the large amount of arithmetical
labour involved in appl 5 dng the ordinary formula may sometimes be
avoided by approximations which are sufficiently accurate for the purpose
in view. We therefore proceed to give a few methods of this kind. They
are not recommended for general use as they will, as a rule, lead to different
results in different hands.

11.2 (1) The means of rows and columns are plotted on a diagram,
and lines fitted to the points by eye, say by shifting about a stretched black
thread until it seems to run as near as may be to all the points. If bi, be
the slopes of these two lines to the vertical and the horizontal respectively,

r=Vb^bz

Hence the value of r may be estimated from any such diagram as fig. 9.8
or 9.9, in the absence of the original table. Further, if a correlation table
be not grouped by equal intervals, it may be difficult to calculate the
product sum, but it may still be possible to plot approximately a diagram
of the two lines of regression, and so determine roughly the value of r.
Similarly, if only the means of two rows and two columns, or of one row and
one column in addition to the means of the two variables, are known, it will
still be possible to estimate the slopes of RR and CC, and hence the correla-
tion coefficient.

(2) The means of one set of arrays only, say the rows, are calculated,
and also the two standard deviations cfx and Oy. The means are then
plotted on a diagram, using the standard deviation of each variable as the
unit of measurement, and a line fitted by eye. The slope of this line to the
vertical is r. If the standard deviations be not used as the units of measure-

253

254

THEORY OF STATISTICS

ment in plotting, the slope of the line to the vertical is rax /ay, and hence
r will be obtained by dividing the slope by the ratio of the standard
deviations.

This method, or some variation of it, is often useful as a makeshift when
the data are too incomplete to permit of the proper calculation of the
correlation, only one line of regression and the ratio of the dispersions of
the two variables being required : the ratio of the quartile deviations, or
other simple measures of dispersion, will serve quite well for rough
purposes in lieu of the ratio of standard deviations. As a special case, we
may note that if the two dispersions are approximately the same, the
slope of RR to the vertical is r.

Plotting the medians of arrays on a diagram with the quartile deviations
as units, 'and measuring the slope of the line, was the method of deter-
mining the correlation coefficient used by Galton, to whom the introduction
of such a coefficient is due.

(3) If 5i be the standard deviation of errors of estimate Uke
we have, from 9.24,

and hence,

But if the dispersions of arrays do not differ largely, and the regression is
nearly linear, the value of Sx may be estimated from the average of the
standard deviations of a few rows, and r determined — or rather estimated
— accordingly. Thus in Table 9.3 the standard deviations of the ten
columns headed 62*5-63*5, 63*5-64*5, etc., are —

2-56

2-26

2-11

2*26

2-55

2*45

2-24

2-33

2-23

2*60

Mean 2-359

The standard deviation of the stature of all sons is 2*75 :
mately

hence approxi-

This is the same as the value found by the product-sum method to the
second decimal place. It would be better to take an average by coimting
the square of each standard deviation once for each observation in the

FURTHER THEORY OF CORRELATION

255

column (or “ weighting it with the number of observations in the column),
but in the present case this would only lead to a very slightly different
result, viz. s=2*362, r=0*512.

Non-linear regression

1L3 We referred in Chapter 9 to the fact that the treatment of cases
when the regression is non-linear is somewhat difficult. We may, by
the methods of Chapter 15, and otherwise, fit curves of any order to the
means of arrays, just as we have fitted straight lines to them ; but the
handling of these regression curves and their interpretation is far more
complicated.

11.4 It is therefore desirable, wherever possible, to deal with variates
which result in linear regression. Now it sometimes happens that if a
relation between X and Y be suggested, we may, either by theory or by
previous experience, throw that relation into the form

Y==^A^B(}>{X)

where A and B are the only unknown constants to be determined. If
a correlation table be then drawn up between Y and (j>{X) instead of Y
and Z, the regression will be approximately linear. Thus in Table 9.5,
page 205, if X be the rate of discount and Y the percentage of reserves
on deposits, a diagram of the curves of regression suggests that the
relation between X and Y is approximately of the form

X{Y--B)=:A

A and B being constants ; that is,

XY=^A+BX

Or, if we make Z Y a new variable, say Z,

Z=A+BX

Hence, if we draw up a new correlation table between X and Z the
regression will probably be much more closely linear.

If the relation between the variables be of the form

Y=.AB^

we have

log Y =log A +X log B

and hence the relation between log Y and X is linear. Similarly, if the
relation be of the form

X”Y^A

we have

log Y =log A --n log Z

256

THEORY OF STATISTICS

and so the relation between log Y and log X is linear. By means of
such artifices for obtaining correlation tables in which the regression is
linear, it may be possible to do a good deal in difficult cases whilst using
elementary methods only.

The correlation ratios

11.5 In view of the importance of linearity of regression it is desirable
to have some criterion which will enable a judgment to be formed whether
a regression is, within the limits permitted by sampling fluctuations,
linear in any given case. We now proceed to discuss a coefficient designed
for this purpose.

Consider a bivariate frequency table, and let spx be the standard
deviation of the pth array of Z's. Let ftp be the number of observations
in this array.

Let

( 11 . 1 )

Then a V is the weighted mean of the variances of arrays, obtained as
suggested in the last sentence of 11.2 (3). Now, let

• • . . ( 11 . 2 )

O^ax

^2^ = 1 .... (11.3)

Then Tfxy is called the correlation ratio of X on Y. Similarly, fjyxt
defined by

is called the correlation ratio of Y on X.

11.6 The correlation ratios may be put in another form, which is much
more convenient for purposes of calculation.

In fact, if Mx is the mean of all the X*s and mpx the mean of an array,
we have, as in equation (6.6),

Nu\='Z{np{s\x + (Mx'-mpx) )

or, using Qmx to denote the standard deviation of mpx, obtained by
“ weighting each mpx according to np, the number of observations in
the array in which it occurs,

+ . . . (11.4)

Hence, substituting in (11.3),

. ( 11 . 5 )

FURTHER THEORY OF CORRELATION

257

The correlation ratio of X on Y is therefore determined when we have
found the standard deviation of X and the standard deviation of the
means of its arrays.

11,7 In 9.22 we saw that

= . . . . ( 11 . 6 )

where is the line of regression of x on y, x and y being the

values of X and Y measured from the mean of the distribution.

Now, for any array for which y is constant,

{x~mpx)

the product term vanishing since 'L{x—mpx)=^0. Hence, summing for all
arrays of y,

But

cr2{j[ —^2

*xy' ax

Hence,

. . . (11.7)

From this we see that 7fxy cannot be less than r in absolute value.

If then

ll{np{mpx--b^Y} =0
ie.

for all arrays. This means that the mean mpx must be on the line of
regression for all arrays, i.e. that the regression is linear.

258

THEORY OF STATISTICS

11.8 The divergence of from therefore measures the departure
of the regression from linearity. It should, however, be noted that
sampling fluctuations may cause Tj^—r^ to deviate from zero even when
the regression is truly linear. We give later a method of testing the
significance of observed fluctuations of this kind.

Calculation of the correlation ratio

11.9 The table on page 259 illustrates the form of the arithmetic

for the calculation of the correlation ratio of son's stature on father's
stature (Table 9.3). In the first column is given the type of the array
(stature of father) ; in the second, the mean stature of sons for that array ;
in the third, the difference of the mean of the array from the mean stature
of all sons. In the fourth column these differences are squared, and in
the sixth they are multiphed by the frequency of the array, two decimal
places only having been retained as sufficient for the present purpose.
The sum-total of the last column divided by the number of observations
(1078) gives o-2„,y=2-058, or awy=l*43. As the standard deviation of
the sons' stature is 2-75 in., ^ya:=0*52. Before taking the differences for
the third column of such a table, it is as weU to check the means of the
arrays by recalculating from them the mean of the whole distribution,
i.e. multiplying each array-mean by its frequency, summing and dividing
by the number of observations. The form of the arithmetic may be
varied, if desired, by working from zero as origin, instead of taking differ-
ences from the true mean. The square of the mean must then be
subtracted from IN to give a^wy.

11.10 If the second correlation ratio for this table be worked out in
the same way, the value will be found to be the same to the second place
of decimals : the two correlation ratios for this table are, therefore, very
nearly identical, and only slightly greater than the correlation coefficient
(0*51). Both regressions, as follows from the last section, are very nearly
linear, a result confirmed by the diagram of the regression lines (fig. 9.8,
page 218). On the other hand, it is evident from fig. 9.10, page 220,
that we should expect the two correlation ratios for Table 9.6 to differ
considerably from each other and from the correlation coefficient.

The student should notice that the correlation ratio only affords a
satisfactory test when the number of observations is sufficiently large for
a grouped correlation table to be formed. In the case of a short series of
observations such as that given in Table 9 7, page 207, the method is
inapplicable.

Rank correlation coefficients

11.11 In calculating the coefficient of correlation from the product-
moment it is necessary that the data should be definitely measured. If
they are not so measured we cannot, in general, determine the coefficient,

FURTHER THEORY OF CORRELATION

259

Example 11.1. — Calculation of the correlation ratio

Sons’s stature on father’s stature

(Data of Table 9 3, page 202)

Type of
array
(Father's
stature)

Mean of
array
(Son’s
stature)

Difference
from mean
of all sons
(68 66)

Square ot
difference

Frequency

Frequency X
(difference)®

64*67

-3*99

15*9201

47-76

65*64

-3 02

9*1204

3-5

31-92

66*34

-2*32

5*3824

43-06

65*56

~3*10

9*6100

163-37

66*68

-1*98

3*9204

33-5

131-33

66*74

~1*92

3*6864

61-5

226-71

67*19

-1*47

2*1609

95-5

206-37

67*61

-1*05

1*1025

142

156-56

67*95

-0*71

0*5041

137-5

69-31

69*07

+0*41

0*1681

154

25-89

69*39

4-0*73

0*5329

141-5

75-41

1 69*74

4-1-08

1*1664

116

135-30

70*50

+ 1*84

3*3856

264-08

70*87

+2*21

4*8841

239-32

72*00

-f3*34

11*1556

28 5

317-93

71*50

+2*84

8-0656

32-26

71*73

+3*07

9-4249

5-5

51-84

Total

1,078

2,218-42

cr2 «2218-42/1078=2‘058 -43

17^=1 -43 /2-75-0-52

though we may sometimes approximate to it by one of the methods of

11 . 2 .

But there may be more smous obstacles than imperfect grouping in
the way of finding the correlation between two variates. In the examples
we have considered up to the present the qualities we have discussed have
been easily measurable, involving such familiar concepts as height, weight,
age and so forth. In certain types of inquiry we may have to deal with
qualities which are not expressible as numbers of units of an objective
kind.

11.12 Consider, for instance, the relation between mathematical and
musical ability in a class of students. “ Ability,” whether of a general
or a specific kind, is a variate in the sense that it varies from one individual
to another ; and it may be a numerical variate if we can decide on some
unequivocal way of measuring it. A very common mode of attempting
to do so is by allotting marks to each student. But such methods are open
to many objections, not the least of which is that diferent examiners would
give difierent marks to the same person. A correlation between the marks
obtained for mathematics and music would, therefore, be likely to depend
to some extent on the examiner, and would not reflect accurately the
relationship between the two qualities.

26 o

THEORY OF STATISTICS

11.13 Difficulties of this type disappear to some extent if we arrange
the students in order of their ability, but do not attempt to assess it
numerically. There will still be some divergence of opinion between
different examiners, perhaps, but it will not as a rule be so serious. We
then allot to each student a number which indicates his position in the
arrangement according to ability, the first being number 1, the second
number 2, and so on. The students are then said to be ranked, and the
number of a particular individual is his rank (cf. 6.33).

11.14 A procedure of this kind is useful in the treatment not only of
data which can be ordered but not exactly measured, but of measurable
data also. For instance, we can easily rank a number of men according
to height without actually measuring them. It is also comparatively easy
to rank a number of shades of a colour, or a number of countries according
to their importance in the export market, where precise numerical measure-
ment would be very troublesome.

In the extreme case we may have situations in which individuals can
be ordered but not measured. Suppose, for example, we have a pack
of cards in which a particular suit, say hearts, is in the correct order
ace, two, . . . king. We then shuffle the pack and examine the order of
the heart cards with the intention of discussing whether the shuffling
process was a good one. The relationship between the orders before and
after shuffling is evidently a possible basis of comparison ; but there is
not even a theoretically measurable variate corresponding to order ” in
this case.

11.15 If we have a set of individuals ranked according to two different
qualities it is natural to inquire whether the ranks can be made to give
us some measure of the degree of relation between the two qualities.

Suppose we have n individuals, whose ranks according to quality A are
Xi, Xg, X3, . . . Xn, and according to quality B are Yg, ^3, . . . Yu,
where the X*s and Y*s are merely permutations of the first n natural
numbers. Let dk=Xk—Yk.

The values of d form a convenient measure of the closeness of the
correspondence between A and B, If all the d's are zero the correspond-
ence is perfect, for an individual whose rank is Xk for A will also be Xh for B.
We cannot, however, take the sum of the d*s as a measure of correspondence,
because that sum is zero ; for the sum of the differences of the X's and Y*s
is the difference of the sums of the X's and the Y's, each of which is the sum
of the first n natural numbers.

A possible measure which suggests itself is the sum of the absolute values
of the d*s, i.e. S j | . This measure and its mean ~ S | f have, in fact, been

used, but like the mean deviation (6.18) they have certain analytical

FURTHER THEORY OF CORRELATION

261

11.16 A more convenient coefficient is obtained as follows —

The values of X range from 1 to n. Their sum is and their

mean is accordingly — This value is also the mean of the Y's.

fi "I" 1

Let us denote by Xk the value of Xk i.e the divergence of Xk

from the mean. Similarly fory*, which we define as Fa—

v'SMS&’j ■ • • • ('1-8)

This is the product-moment coefficient of correlation between X and Y.
We shall call p Spearman's rank correlation coefficient. It may be
expressed very simply in terms of n and the i's.

For, as we saw in 6.15,

Now,

Hence,

and substituting in

E(i2)=2(ZA-YA)2=S(x->^)2

^X{x^)+i:{y^)^2^xy)

I.(xy)=i

n^—n

( 11 . 8 )-

p=l

n^—n

Z(d^)

(11.9)

Example 11.2. — ^The rankings of ten students in mathematics and
music are as follows —

Mathematics : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Music : 6, 5, 1, 4, 2, 7, 8. 10, 3, 9

What is the coefficient of rank correlation ?

The differences d are (mathematical rank minus musical rank)

-5, -3, +2, 0, +3, ~1, -1, -2, +6. +1

These add to zero, as they should.

The squares of d are

25, 9, 4, 0, 9, 1, 1, 4, 36, 1

P=:=l

540

990

=:-f0-45

which add up to 90,
Hence, from (11.9),

262

THEORY OF STATISTICS

11.17 The rank correlation coefficient varies from +1 to ~1. If the
rank correlation is perfect, all the i's are zero. If, on the other hand, the
ranks are such that the first, second, third in one order correspond to the
«th, (w— l)th, 2)th, ... in the other, /> =— 1. The proof is slightly
different according to whether n is even or odd. If it is odd, say =2w+l ,
the d's are

and

2m, 2m —2, ... 2, 0, ~2, . . . —(2m— 2),

S(^f2)=2{(2m)2+(2m-2)2+ . . +2^}

__8m(m+l)(2m+l)

-2m

Hence,

p=:l-

8m(m-|-l)(2m+l)

(2m+l){(2m-i-l)2~l}

If n is even, say =2m,

i:(t/2)=2{(2m-l)2+ . . . 4-12}

and

=^(4»l2-l)

1 as before.^

11.18 A second rank correlation coefficient which has certain advantages
over Spearman's may be obtained as follows : Consider again the data
of Example 11.2, and consider the order of each possible pair in the two
rankings. If any pair is in the same order in both we allot it the score
4“1, if in the opposite order the score —1. For instance, of the pairs
65, 61, 64, 62, 67 the first four are in the reverse order in the second
ranking as compared with the first and each scores —1 ; the fifth, 67, is
in the same order and hence scores -|-1 ; and so on. There are ^®C2=45
possible pairs. The maximum score, if both rankings are the same, is
45. The minimum score, if one is the inverse of the other, is —45. In
our present example the total score will be found to be 15. We then
define a rank correlation coefficient r as

Score

Maximum possible score

iThe property of varying between -J- 1 and — 1 does not belong to a similar coefficient

proposed by Spearman, and known as bis “foot-rule,** viz. jg—

It may be shown in the above manner that R varies from —0*5 to -hb and for this
reason alone R seems an undesirable coefficient.

FURTHER THEORY OF CORRELATION

263

11.19 Generally, if S is the score in a ranking of n we have

^ S

( 11 . 10 )

T may also be regarded, in a sense, as a product-moment correlation.
Suppose that for any two ranks i, j, we allot the value +1 if i > j and
~1 if i < y. Call this value a*;, so that

(i%j —

-1

i > j
i < j ^

Similarly let represent a corresponding quantity in the second ranking.
We then have

r nini

V^(a%) S(&V) ■ ■ ■ ' ^ ^

for is merely the number of possible pairs \n{n—\) and the

numerator is the score S as defined above.

Example 11.3. — A set of 15 recruits are given a preliminary test to
admit them to a course of training and, after the completion of training,
a proficiency test. Their ranks are —

Candidate . .ABCDEFGHIJ KLMNO

Rank (prelim.) 7 4 1 3 14 13 10 12 5 9 8 2 11 15 6

Rank (profic.) . 4 6 3 7 15 11 14 12 1 13 5 2 9 10 8

Does this suggest that the preliminary test was a good predictor

of the results in the proficiency test ?

To calculate r it is convenient to rearrange one ranking so as to be in
the natural order 1, . . . «. If we do so for the ranking in the preliminary
score we have, for the ranking in the proficiency score —

3 2 7 6 1 8 4 5 13 14 9 12 11 15 10 ... (a)

The score obtained by considering the first member 3, in conjunction with
the others is 12—2 = 10, for there must be 12 members greater than 3
and 2 less than it. Similarly the score (apart from that involving the 3
which has already been counted) involving the 2 is found to be 11. That
involving the 7 is 4. The total score (the reader should check this result)
is then

10+1 1+4+5+10+5+8+7-2-3+4-1 -hO-1 =57

264 THEORY OF STATISTICS

Thus, since the maximum possible score is 105 we have

indicating a moderate, but not a very high, correlation between the
rankings in the two tests.

When one ranking is in the natural order a slightly simpler method of
calculating t may be used. In the ranking {a) we count the number
of members greater than 3 lying to the right of 3 (giving 12), then the
number greater than 2 lying to the right of 2 (again 12) and so on. If R
is the total score so obtained

• • • • ( 11 - 12 )

a relation which the reader can easily prove for himself.

11.20 It is useM to remember that for large n the following relation
usually holds approximately except for values of /> or r near to unity —

(11.13)

For instance, in the data of Example 11,2 we found /x=0*45 and t=0*33.

11J21 It is rather more troublesome to calculate r than to calculate p,
but r has advantages for more advanced work.

[а) Where sampling effects are in question the significance of r may
be tested by known methods but little is known about p except in one
special case (cf. 19.31-19.34).

(б) T may be extended to partial rank correlations.

(c) If an extra member is added to the ranking (as, for instance, if one
has been accidentally omitted or further information arrives late) it is
easier to recalculate r than p. In fact, in making a new determination of
p, it may be necessary to re-rank many of the members and hence to
recalculate the values of d ; whereas for r we need only consider the
additional scores attaching to the new member added.

Tied ranks

11.22 In some classes of ranking work, as for instance in arranging
students in order of merit, it is impossible to distinguish between a number
of adjacent individuals. In such a case it is customary to average the
ranks and to assign the same rank to each even though it may be fractional.

FURTHER THEORY OF CORRELATION

265

For example, in a ranking of 10, we may be able to assign one individual
to the rank 1, but be unable to decide which of the next two members
shall be second and which third. They are therefore " tied and each is
given the rank |(2+3)=2|. The next member is then ranked 4, and so
on. If we had to tie the next three members we should allot to each
the rank |(4+5+6)=5. The general procedure will now be clear.

11.23 When ranks are tied we have a choice in the calculation of p and r.
Let us in the first place determine the effect on the sum of squares of the
ranks of tying t individuals occupying the ranks ^+1, k+2, , , , k-\~t
The sum of squares of untied ranks is —

. . . (ft “h^^(^~t"l)(2^'i“l)

The sum of squares of the tied ranks is —

The difference is then —

4. 1 ) (2/ + 1 ) — + 1 ) 2 (^3 —

Consequently, if we tie t ranks the sum of squares is lowered by
The mean value of the ranks is the same, |(w+l) and hence the variance
of the tied ranking is lowered by Moreover, the effect of

tying different sets is evidently additive, so that if we have a ranking
with ties of • ti and

j=i

the variance of the ranking is —

Similarly it will be found that

. (11.14)

. (11.15)

where Ty is the quantity corresponding to Tx for the second ranking.

Hence, if we continue to regard p as the product-moment correlation
of the rankings we have —

-2Tx}i{U»’‘-^) -2Ty}i

. (11.16)

as compared with the simple formula (11.9) to which it reduces if

Tj,==Ty=0.

266

THEORY OF STATISTICS

11.24 The reader will sometimes find other formulae in use. For instance,
(11.9) is sometimes used as it stands for tied ranks. This is certainly
wrong. An alternative is to convert ^{xy) for ties as in (11.15) but not
to correct the variances, which leads to the formula

' * o

n^—n

(11.17)

to which (11.16) reduces if we put Tx = Ty=--0 in the denominator only.

11.25 From some points of view (11.17) may be justifiable. Suppose
we have two judges who rank a number of candidates identically, though
there are ties present. In such a case (1 1 .16) is the form to use, for we are
measuring the agreement between them and the correlation should be
unity. Both judges may be wrong, but that is not the point. We are
measuring their agreement, not their accuracy.

But if we have one observer ranking a number of objects which really
have an objective order (11.17) may be preferable. The observer may tie
certain ranks because of an inability to distinguish between the individuals
concerned. In using (11.17) we take this into account in ascertaining the
covariance of (11.15) : but in deciding to make allowance in the variance
we are refusing, so to speak, to give him credit for clustering his values
because he ought not to do so, there being a really objective order. The
effect of using (11.17) instead of (11.16), of course, is to give a lower value
to p, which appears to conform to the common-sense requirements of the
position wherein we are measuring the observer's ability to rank individuals
in their real order,

11.26 In the calculation of r we allot to any tied pair the score 0, this

being the intermediate point between the scores of +1 ~1 which

would result if one were greater than the other. The effect of this is
to lower the maximum possible score for X by

the summation taking place over the ties as for
(11.16) we shall then have

. (11.18)
Corresponding to

(11.19)

and corresponding to (11.17)

\n{n — \)

( 11 . 20 )

In both these formulae the score S is, of course, affected by ties.

FURTHER THEORY OF CORRELATION 267

Example 11.4. — Two foremen rank ten employees according to suitability
for promotion as follows —

Employee . .ABCDEFGHIJ

Foreman 1. .1|1|3 4 66 6 8 9J9J

Foreman 2. .1 24446789 10

In the first ranking there are three sets of ties and we have —

Similarly

ry=A(33-3)

= 2

The differences d are

-1, 0, 2. 0. -1. 0, i

and hence

S{(f®)=7

Hence from (11.16)

165 - 3 - 2-7

^ V(159x161)

=0-956

The scores S contributing to r, taking the first emplo^^ee A with the
others, then B with C . . . J and so on, will be found to be

8 “|“8 “{“ 5 -|“ 0 ’ F 3 “ f ‘3 - i ~3 ~ f '2 -[-0 =37

We also have

Hence, from (11.19^

^a’=4{-+3.2+2}

?7y=3

''“V{40x4^

=0-903

Either coefficient indicates a high degree of agieement between the judges.

268

THEORY OF STATISTICS

Relationship between rank correlation and product-moment correlation

11.27 The rank correlation coefficients as we have introduced them are
merely measures like the coefficients of association, contingency and
product-moment correlation, of the correspondence between two quantities.
Like those coefficients, they are affected by sampling fluctuations.

They are, however, more easily calculated than most coefficients, and for
this reason some writers have advocated their use as a substitute for the
product-moment coefficient between the actual measurements, and for
estimating the product-moment coefficient from a normal population. We
proceed to examine this practice briefly.

Grade correlation

11.28 We referred at the end of Chapter 6 to such quantities as quartiles,
deciles and percentiles, which are values of the variate dividing the total
frequency into certain specified proportions. For instance, the seventh
decile is the variate value such that seven-tenths of the distribution lie
below it, i.e. exhibit values of the variate less than the decile.

Generally, we may regard the grade of an individual as the proportion
of individuals which lie below him (cf. 6.31). If the population i^ con-
tinuous, the range of grades will also be continuous.

11.29 To each individual in a bivariate population there will be attached
two grade numbers, one for each variate, and if the population is correlated
the grades will also be correlated. In fact, it has been shovm that if the
population is normal, pg, the grade correlation, and r, the ordinary correla-
tion (both calculated by the product-moment method), are related by the
equation

.... ( 11 . 21 )

11.30 Ranks and grades are connected by a simple relation. In fact,
if an individual is of rank k, there are ^—1 individuals below him (assuming
that the ranking proceeds from the lowest variate value). If we admit,
conventionally, that one-half of the individual is to be regarded as lying
to the left of the line of division which he makes, and one-half to the
right, his grade, gk, is given by

gk^{k-l)+i=k-i . . . ( 11 . 22 )

It follows that the correlation between ranks is the same as the correla-
tion between grades. But in a population which is finite and discontinuous
(and ranking is in practice applied to comparatively small populations of
twenty or thirty individuals) it does not follow that

r=2sin(^)

. (11.23)

FURTHER THEORY OF CORRELATION

269

Equation (11.21) was obtained by considering grades in a continuous
population, and equation (11.23) is at best an approximation, depending on
assumptions which are often of doubtful legitimacy. This is a fact which
has not always been appreciated. We may, perhaps, clarify the point by
considering the data of Example 11.2.

Example 11.5. — In Example 11.2 we found —

p = -f0*45

If we apply -(11.23) we find —

^ f=2sinl3-5°

==+0-47

Let us consider what this means.

The value r purports to be a correlation coefficient such as would have
been obtained by the product-moment method if the two variates had been
measurable in the ordinary way. Let us, for the sake of argument, agree
that mathematical and musical abilities are capable of measurement.

Now there are only ten members in this population, and it cannot be
regarded with any degree of accuracy as a continuous normal papulation.
The use of (1 1,23) in finding the correlation in the population of ten is there-
fore of doubtful validity, to say the least.

But it is possible to look at this from rather a difierent point of view,
and to regard the ten students as a sample from a practically infinite
population which ts continuous and normal. The value r is then taken to
be an estimate of the correlation coefficient in this population.

The legitimacy of this procedure will depend on the extent to which the
grade correlation in the sample can be taken to represent the grade correla-
tion in the population. It will, we think, be sufficiently evident from the
smallness of the sample that the two are likely to diverge considerably
owing to sampling fluctuations.

Furthermore, in the comparatively small samples to which (11.23) is
applied — the labour of calculating the rank correlation coefficient for large
samples is very tedious — it is difficult to obtain any satisfactory e\ddence
from the data themselves that the population can properly be regarded as
normal ; and even if the distribution of each of the variates, taken singly,
can be rendered normal by some appropriate transformation of the variate
which squeezes or stretches the scale of measurement, it does not
necessarily follow that the correlation distribution can in this way be
rendered normal.

As a matter of interest we may record that, corresponding to (il.lG)
for p we have also the relation

. TTT

. ( 11 . 24 )

270

THEORY OF STATISTICS

The use of this equation is, of course, subject to the same objections as
lie against (11.23).

Use of (11.23) and (11.24) should therefore be made with the utmost
reserve. It would probably be better to avoid them altogether and rely
on the rank correlation coefficient.

11.31 The relationship between the product-moment coefficient and
the rank correlation coefficients might profitably be subjected to further
investigation, particularly for small numbers of individuals. As we have
just seen, with the present state of our knowledge, the use of the rank
coefficient is not to be recommended as a brief method of estimating the
product-moment coefficient. It is, however, of service as a quick method
of gauging relations between variates w-hich are not normally distributed
and in any case it is useful where the variates can be ranked but not
measured for either practical or theoretical reasons.*

Tetrachoric r

11.32 To complete our account of methods which have been devised
as alternatives to the use of the product-moment correlation coefficient in
cases where, for some reason, that coefficient cannot be computed, we may
refer to a process specially adapted to the 2 >< 2 contingency table.

Consider such a table in the schematic form —

Not-^

Total

B .

Clr-\‘b

Not-B

c-\-d

Total

a+c

h-^d

Let us assume that our attributes A and B are, in theory, based on
measurable quantities ; and let us suppose further that the population
would be normally distributed with respect to those quantities as variates.
Then we may regard the above table as the result obtained by dividing a
bivariate normal population into four sections, a division of the Z-variate
at some point, say h, and a division of the Y-variate at some point k. If
we picture the population as a solid figure, as in fig. 9.1, page 208, the
frequencies 6, c and i will be the volumes into which the population is
divided by planes perpendicular to the X and Y axes through the points
Z=^ and Y=^, respectively.

The problem then arises, given a, b, c and d, what are the values of
h and k (in terms of the standard deviations of X and Y), and what is
the value of r ?

* For some further de\ eiopments ot this subject see Kendall’s Ranh Correlation Methods,
Second edition, 1955

FURTHER THEORY OF CORRELATION

271

11.33 A discussion of this problem, which involves some difficult mathe-
matics, is outside the scope of this book. The student may be referred
to Kendall’s Advanced Statistics, vol. 1, for an account of the method and
to Tables for Statisticians and Biomeiricians, Parts 1 and II, for tables
which are almost indispensable in working out r for any given case.

A value of r obtained in this way is said to be tetrachor'ic.

The coefficient has often been used to obtain a value of the correlation
(so-called) for a contingency table, usmg some reduction to the four-fold
form by amalgamating adjacent arrays, or possibly making more than one
such reduction and averaging the results. As such tables are very often
far from normal, it is always desirable to test the normality by using more
than one reduction. In any case the reader should be informed precisely
as to the reduction used.

The product-moment correlation coefficient for a 2x2 table

11.34 The correlation coefficient is in general only calculated for a table
with a considerable number of rows and columns, such as those given 1 x 1
Chapter 9. In some cases, however, a theoretical value is obtainable
for the coefficient, which holds good even for the limiting case when
there are only two values possible for each variable (e.g. 0 and 1) and
consequently two rows and two columns (cf. Exercises 11.5 and 11.6).
It is therefore of some interest to obtain an expression for the coefficient
in this case in terms of the class-frequencies.

Using the notation of Chapters 1-3 the table may be written in the
form —

Values of
second
variable

Values of first
variable
^'1

Total

(AB)

iaB)

(B)

(ay?)

(A)

Total

(A)

(a)

Taking the centre of the table as arbitrary origin and the class-interval,
as usual, as the unit, the co-ordinates of the mean are —

The standard deviations a,, are given by

a,2=0-25-|®=(/i)(a)/iV2

C7*8=O-25-^>*=(B)0ff)/2V*

THEORY OF STATISTICS

Finally,

Writing

Y.{xy) =i{ (AB) +(ajS)-(Aj&)-(ccB) } -Nl^
(AB)-{A){B) IN=d

(as in Chapter 2) and replacing g, ^ by their values, this reduces to
Whence

"“VPTRIW) ■

We may also put this in the form

. (11.25)

.... (11.26)

where is the square contingency as defined in 3.8.

This value of r can be used as a coefficient of association, but, unlike
the association coefficient of Chapter 2, which is unity if either {AB)^{A)
or [AB)={B), r only becomes unity if (2l5)=(^)=(B). This is the
only case in which both frequencies {aB) and [Afi) can vanish so that
[AB] and {a^) correspond to the frequencies of two points,
on a line. Ob\dously this alone renders the numerical values of the two
coefficients quite incomparable with each other. But further, while the
association coefficient is the same for all tables derived from one another
by multipl 5 dng rows or columns by arbitrary coefficients, the correlation
coefficient (11.25) is greatest when {i4)=(a) and (B) =(;/?), i.e. when the
table is symmetrical, and its value is lowered when the symmetrical
table is rendered as 5 nnmetrical by increasing or reducing the number of
id's or B's. For moderate degrees of association, the association coefficient
gives much the larger values. The two coefficients possess, in fact,
essentially different properties, and are different measures of association
in the same sense that the geometric and arithmetic means are different
forms of average, or the semi-interquartile range and the standard devia-
tion different measures of dispersion.

H.35 The student should realise that the product-sum correlation
and the tetrachoric correlation are also two entirely different measures
with quite different properties. The one is in no sense an approximation
to the other, and the two may often differ largely.

Intraclass correlation

11.36 We have previously considered correlations between two definite
defined variates, such as age and yield of milk in cows, or stature of
father and stature of son ; but there occurs, mainly in biological studies,

FURTHER THEORY OF CORRELATION

^73

a rather different kind of correlation which we will now proceed to discuss.

Suppose we are examining the relationship between the heights of
brothers, and consider a pair of brothers. Our two variates will be (1)
the height of the first brother, and (2) the height of the second brother.
The question is, which are we to regard as the first brother and which as
the second? It is not diflicult to lay down rules which would enable us
to make a distinction — ^for instance, we might take the elder brother
first, or the taller brother first. But if we did this and drew up a correla-
tion table for all such pairs, we should not be answering the question
as to the relation between brothers in general, for we should only get a
correlation between the height of taller brothers and that of shorter
brothers, or the height of elder brothers and the height of younger brothers.

11.37 The relationship of brotherhood is in fact symmetrical ; if is
the brother of B, then B is the brother of A. When we are considering
only the relationship in height implied by relationship of blood, there is
no relevant character to enable us to single out one brother as the first.

We accordingly treat the problem by taking each pair of brothers in
two ways : (1) with the height of A as the first variate and that of B as
the second, and (2) with the height of B as the first variate and that of
A as the second. Similarly, if there are k brothers in the family, we enter
in the correlation table the results of taking pairs in all possible ways,
which number 1). For example, if we have a family containing
three brothers with heights 5 ft. 9 in., 5 ft. 10 in. and 5 ft. 11 in., they
may be regarded as giving six pairs of variate values —

5 ft. 9 in. with 5 ft. 10 in, 5 ft. 10 in. with 5 ft. 9 in.

5 ft. 9 in. with 5 ft. 11 in. 5 ft. 11 in. with 5 ft. 9 in.

5 ft. 10 in. with 5 ft. 11 in, 5 ft. 11 in. with 5 ft. 10 in.

11.38 Generally, if we have n families, each with k members, there will
be nk{k~-\) pairs, and hence the same number of entries in the table.

Such a table is called an intraclass correlation table, and the correlation
between the two variates is called intraclass correlation.

Tables in which all the families have the same number are of particular
importance, and we will consider them first. It is, however, permissible
to apply the term intraclass correlation to the symmetrical table derived
from families which have different numbers of members. This case we
shall consider in 11.42.

11.39 The intraclass correlation table has certain peculiarities, and is
not of such a general type as the ordinary table which we have considered
hitherto (and which, for the purposes of distinction, is sometimes called
an inter class table).

Let the variate values in the first family be

274

THEORY OF STATISTICS

those in the second family being

^21 ^22 • • •

and so on, those in the wth family being

^n\ ^n2 • • • ^nk

Consider the mean of the X- variate.

In the table the value will be associated as an Z-variate with each
of the (^—1) values x^2 • • • Hence it appears (^—1) times. Similarly,
every other value appears 1) times. Hence the sum of the marginal
row, corresponding to the Z- variate, is 1 )S(a;), the summation ex-
tending over all values. But there are nk{k--\) members in the table.
Hence,

( 11 - 2 ^

Similarly,

( 11 . 28 )

i.e. the means of the variates are the same. This must evidently be the
case, for the table is symmetrical.

For the variance of Z we have —

and since each x—X occurs times,

.... ( 11 . 29 )

the summation, as before, extending over all the values of x.

Similarly,

We therefore write

a=o’x==crj.

FURTHER THEORY OF CORRELATION

275

11,40 For the correlation coefficient r we have

• ■ • ( 11 - 30 )

where the summation S' extends over all the possible pairs.

We can put this formula into a much simpler form.

Consider the terms in (11.30) for which the first term is They

will be the (^—1) terms of the following series —

(^n +• • • • +(%i

= (^ll-^){(%2+^13+ - • •

Now write

-^i=^(%i+%2+ • • * +^ifc) • • • (11-31)

i.e. is the mean of the members of the first family. Then our expression
becomes

The sum S' of (11.30) will contain nk such terms.

Hence,

. . (11.32)

the summation extending over all the nk members.

Now,

yfeS(Xi-X)(%i-X)

=sum of n terms like k x

21" extending over the n families ; and

2:(A;u-~X)2=«to2

Hence, from (11.32),

nk{k-l)G^r==k^i:"(X^-X)^-Ghtk

Now is the variance of the means of families about the

mean of the whole. Calling this we have

nk{k--l)a^r=kH(Jm^—oHk

{l+r{k--l)}G^=::kGm^ . . . (11.33)

276

THEORY OF STATISTICS

This result gives us the intraclass correlation in terms of the variance of
the distribution (according to either variate) and the variance of the
means of families.

Example 11.6. — In five families of 3 the heights of brothers are : 5' 9"",
5' 10", 5' 11" ; 5' 10", 5' 11", 6' 0" ; 5' 11", & 0", 6' 1" ; 6' 0", 6' 1", 6' 2" ;
6' r, & 2" 6' 3". Find the intraclass coefficient of correlation.

Here the mean of the whole =6'.

«y®=5^{9+4+l +4+1 +1 +1 +1 +4+1 +4+9}

^ 40^8
15 3

a«2=|{4+l+0+l+4}=2
Hence, from (11.33),

{l+2r}|=3x2
1 +2^=2 -25

+0-625

11.41 We may notice two rather unusual results which follow from
equation (11.33).

In the first place, since Un? is not negative,

l+r(ib-l)^0

and hence,

Thus, whereas the interclass correlation coefficient can vary from —1 to

+1, the intraclass coefficient cannot be less than — ^ For example, in

families of threes the intraclass coefficient cannot be less than —
Secondly, let us consider the correlation within a single family, i.e. when
«= 1 .

In this case, am^=0, and hence

For ft =2, 3, 4, . . . this gives the successive values of r= —1, —
—J, ... It is clear that the first value is correct, for the two values
and x^ determine only two points and and the slope of the line
joining them is negative.

The student should notice that a corresponding pegative association
win arise between the first and second members of the pair if all possible

FURTHER THEORY OF CORRELATION ^77

pairs are chosen from a population in which the variates can assume only
two values, say 0 and 1, or in which only and not-^’s are distinguished.
We use this result later in 17.36.

11.42. Reverting now to the more general case, suppose we have n
families whose members number Ag, . . . kn.

The ith family contributes 1) pairs to the intraclass table, and

hence the total number of pairs is l)}=iV, say, the summation

extending over the n families.

Let the variate values be

^12 - •

^21 ^22 • •

• ^2lZ

^nl • •

• ^nkn

As in 11.41, we see that in the intraclass table each member of the first
family appears 1) times, each of the second (^ 2 “^) tiroes, and so on.
Hence,

. . . (11.34)

the summation S' being carried over all members of the ith family and S
over ail families.

Similarly,

a/=c7r^=^'{(*.-l)S(;c.y-X)2} . . (11.35)

and

the summation extending over ail possible pairs,
and this, as in 11.40, reduces to

These formulae are considerably more complex than those of 11.40,
but reduce to those forms if ki, is constant for ail families.

SUMMARY

1, In cases where the data are incomplete, or in order to avoid lengthy
calculation, it is possible to use various methods of approximating to the
product-moment coefficient of correlation, provided that the regression is
approximately linear.

2. Cases in which the regression is non-linear can sometimes be reduced
to the linear case by a suitable transformation of the variates.

278

THEORY OF STATISTICS

3. The correlation ratio of X on Y is given by

where a| is the variance of X, is the weighted average of the variances
of arrays and the variance of the means of X-arrays, weighted
according to the number of individuals in the arrays.

4. Vxy—^^ cannot be negative, and if it is zero the regression of X on Y
is linear.

5. Spearman's rank correlation coefficient is given by

__L(pcy)__

where x and y are the deviations of the ranks X and Y from the mean

n+l.

6. If

dk=^(Xk-Yk)

/0 = 1 —

^ fr—n

7. The rank correlation coefficient t is given by

^ 2R

where 5 is the sum of scores obtained by allocating +1 if pairs of ranks
are in the same order in the two rankings and —1 in the contrary case ;
and R is the sum of scores for positive scores only.

8. The coefficient of intraclass correlation is given by

{l+Kft-l)}cr2=W

where a is the standard deviation of X and Y, and Om is the standard
deviation of the means of families, there being n families each of k
members.

EXERCISES

11.1 Find to 3 places of decimals the correlation ratio of X on Y and of
y on X for the distribution of cows of Table 9.4, page 204 +0*219).

Hence, show that

^|^-“^^= 0-023

FURTHER THEORY OF CORRELATION 279

11.2 Find the correlation ratios of the distribution of mamages of Table

9.2.

11.3 In a test of ability to distinguish shades of colour, 15 discs of
various shades, whose true orders are 1, 2, ... 15, are arranged by a subject
in the order 7, 4, 2, 3, 1, 10, 6, 8, 9, 5, 11, 15, 14, 12, 13. Find the rank
correlation coefficients p and r between the real and the observed ranks.

11.4 Ten competitors in a beauty contest are ranked by three judges
in the orders

1, 6, 5, 10, 3, 2, 4, 9, 7, 8

3, 5, 8, 4, 7, 10, 2, 1, 6, 9

6, 4, 9, 8, 1, 2, 3, 10, 5, 7

Use rank correlation coefficients to discuss which pair of judges has the
nearest approach to common tastes m beauty.

1 1.5 (Cf. Pearson, “ On a Generalised Theory of Alternative Inheritance,’'
PML Trans., A, 1904, 203, 53.) If we consider the correlation between
number of recessive couplets in parent and in offspring, in a Mendelian
population breeding at random (such as would ultimately result from an
initial cross between a pure dominant and a pure recessive), the correlation
is found to be 1 /3 for a total number of couplets 7i. If w=l, the only
possible numbers of recessive couplets are 0 and 1, and the correlation
table between parent and offspring reduces to the form

Offspring

Parent

0 1

Total

Verify the correlation, and work out the association coefficient Q.

11.6 (Cf. the above, and also Snow, Proc. Roy. Soc., B, 1910, 83, 42.)
For a similar population the correlation between brothers, assuming a
practically infinite size of family, is 5/12. The table is

Second

First brother

brother

Total

Verify the correlation, and work out the association coefficient 0,

28 o

THEORY OF STATISTICS

11.7 Establish equation (11.26).

7TX

11.8 Show by drawing a graph that the values of x and 2 sin— are

never very different for the range — 1 < a; < 1 and that the greatest difference
is about 0-018 (Cf. equation (11.23)).

11.9 Referring to the notation of 11.34, show that we have the following
expressions for the regressions in a fourfold table — -

0-1 Nd {AB) (AS)

(B) (A)

02 _ NS _{AB) (aB)

\ {A){a) (^) (a)

Verify on the tables of Exercises 11.5 and 11.6.

11.10 In four pea-pods, each containing eight peas, the weights of the
peas are, in hundredths of a gramme : 43, 46, 48, 42, 50, 45, 45 and 49 ;
33, 34, 37, 39, 32, 35, 37 and 41 ; 56, 52, 50, 51, 54, 52, 49 and 52 ; 36,
37, 38, 40, 40, 41, 44 and 44. Find the coefficient of intraclass correlation.

11.11 (Data from O.H. Latter, Biomefrika, 1905, 4, 363.)

The following table shows the length of cuckoos' eggs fostered by
various birds —

Foster parent

Length of egg (units J millimetre)

41 42 43 44 45 46 47 48 49

Totals

Robin

11 2

Wren

— —

—

Hedge-sparrow

—

5 —

Totals

16 2

188

Find the coefficient of intraclass correlation, and state how many entries
there would be in the intraclass correlation table.

11.12 lit consecutive ranks are replaced by a single tie, show that, for
both p and r, the resulting coefficients are the means of the 1 1 coefficients
obtained by permuting the t original ranks in all possible ways. Show
that this remains true if there are several sets of tied ranks in either
ranking.

CHAPTER TWELVE

PARTIAL CORRELATION

Mutiple correlation

12.1 In Chapters 9 to 11 we developed the theory of the correlation
between a single pair of variables. But in the case of statistics of
attributes we found it necessary to proceed from the theory of simple
association for a single pair of attributes to the theory of association for
several attributes, in order to be able to deal with the complex causation
characteristic of statistics ; and similarly the student will find it impossible
to advance very far in the discussion of many problems in correlation
without some knowledge of the theory of multiple con elation, or correlation
between several variables.

For example, in considering the relationship between the number of
children per family, level of income and age at marriage, it might be
found that the number of children was negatively correlated with income
and also with age at marriage ; and the question might arise how far
the first correlation was affected by the fact that people with higher
incomes tend to marry later. The question could not at the present
stage be answered by working out the correlation coefficient between the
last pair of variables, for we have as yet no guide as to how far a correlation
between the variables 1 and 2 can be accounted for by correlations between
1 and 3 and 2 and 3.

Again, a marked positive correlation might be observed between, say,
the bulk of a crop and the rainfall during a certain period, and practically
no correlation between the crop and the accumulated temperature during
the same period ; and the question might arise whether the last result
might not be due merely to a negative correlation between rain and
accumulated temperature, the crop being favourably affected by an
increase of accumulated temperature ij other things were equal, but failing
as a rule to obtain this benefit owing to the concomitant deficiency of rain.
In the problem of inheritance in a population, the corresponding problem
is of great importance, as already indicated in Chapter 2. It is essential
for the discussion of possible hypotheses to know whether an observed
correlation between, say, grandson and grandparent can or cannot be
accounted for solely by observed correlations between grandson and
parent, parent and grandparent.

Partia! r^ressiom and correlation coeMdents

12.2 Problems of this type, in which it is necessary to consider simul-

281

282

THEORY OF STATISTICS

taneously the relations between at least three variables, and possibly
more, may be treated by a simple and natural extension of the method
used in the case of two vanables. The latter case was discussed by form-
ing linear equations between the two variables, assigning such values
to the constants as to make the sum of the squares of the errors of estimate
as low as possible : the more complicated case may be discussed by
forming linear equations between any one of the n variables involved,
taking each in turn, and the n—l others, again assigning such values to
the constants as to make the sum of the squares of the errors of estimate
a minimum. If the variables are X^, . . . the equation will

be of the form

X-^=a-\-h2X2+h^X^+ . . . -^h^X^

If in such a generalised regression equation we find a sensible positive
value for any one coefficient such as we know that there must be a
positive correlation between X^ and X^ that cannot be accounted for by
mere correlations of X^ and X^ with Z 3 , X^ or X^, for the effects of
changes in these variables are allowed for in the remaining terms on the
right. The magnitude of gives, in fact, the mean change in X^
associated with a unit change in X^ when all the remaining variables are
kept constant.

The correlation between X-^ and Xg indicated by may be termed
a partial correlation, as corresponding with the partial association of
Chapter 2 , and it is required to deduce from the values of the coefficients
6 , which may be termed partial regressions, partial coefficients of correlation
giving the correlation between X^ and X^ or other pair of variables when
the remaining variables X 3 . . . X„ are kept constant, or when changes
in these variables are corrected or allowed for, so far as this may be done
with a linear equation. For examples of such generalised regression
equations the student may turn to the illustrations worked out later
in this chapter.

12.3 With this explanatory introduction, we may now proceed to the
algebraic theory of such generalised regression equations and of multiple
correlation in general. It will first, however, be as well to revert briefly
to the case of two variables. In Chapter 9, to obtain the greatest possible
simplicity of treatment, the value of the coefficient r^p /o-^o-g was deduced
on the special assumption that the means of all arrays were strictly
collinear, and the meaning of the coefficient in the more general case was
subsequently investigated. Such a process is not conveniently applicable
when a number of variables are to be taken into account, and the problem
has to be faced directly ; i.e. required, to determine the coefficients and
constant term, if any, in a regression equation, so as to make the sum of
the squares of the errors of estimate a minimum.

12.4 To solve this problem we proceed as in 9.20.

PARTIAL CORRELATION 283

Let us measure the variates from their respective means,

denoting the quantities so obtained by
Then the regression equation of, say, on X 2 . . . may be written
in the form

Xi~a^-\-h2X^-\-h^x^-\-‘ . . .

We have to find h^, , , , such that

Ei==^{Xi dy—h^x^—' . . . — b^Xji)^

is a minimum, the summation taking place over all sets of values of

Xy , . , Xn*

Now,

Ey=E{ay^)-\-ll{Xy ^>2^2 • • •

the product term

2 i:{ai{xi-b^i- . . . -b„x„)}

vanishing, since Xy, etc. are measured from the mean.

Hence we have, for the minimum value of Ey,

ay===0

Now, if 62 is chosen so that Ey is a minimum, the value of Ey, when
(b^+S) is substituted for is increased no matter how small d may be ;

i.e.

S{^1-“(^2+^)^2~' • • • • • • -^n) ^

Expanding the left-hand side, and neglecting S^, which can be made as
small as we please compared with d,

Xix^-^b^x^— . . . -b„x^)^-21>{x2{Xy-b2X2- . . . — Mn)}^

'^^{Xy ^2^2 • • •

S {^2(^1— ^2^2— • • •

Now this is to be true for all small values of d, positive or negative.
If l,{x2(Xy—b2X2^ • . . Were not zero, this would be impossible,

for if it were positive, say, we could take d positive and the inequality
would not be satisfied.

Hence,

S {^2(^1 ‘“^2^2— . . . —

Similarly, considering 63 instead of b^, we have

. . . -^> a )}=0

and so on, there being {w— 1) equations. These are sufficient to determine
the (w— 1) quantities • • • b^, and hence our problem is solved.

284

THEORY OF STATISTICS

Notation

12.5 At this point we introduce a flexible notation which will enable
us to consider any regression equation.

We write —

34 . . . «^ 2+^3 3 24 . . . 23 , (n-l)^n (12.1)

The quantities h are partial regression coefficients. The first subscript
attached to the h is the subscript of the letter on the left (the dependent
variable). The second subscript is that of the x to which it is attached.
These are called primary subscripts.

After the primary subscripts, and separated from them by a point,
are placed the subscripts of the remaining variables on the right. These
are called secondary suh^npts.

Equation (12.1) is the regression equation of x-^. Similarly, in accord-
ance with the rules we have just laid down, we have —

^2—^21.84 . . . n%+^23.14 . . . • • • +^2n.l8 . . (n-l)^n

and so on.

It should be noted that the order in which the secondary subscripts are
written is immaterial ; but this is not true of the primary subscripts ; e.g.
^ 12 . 3 . , .n 3-nd ^? 2 i .3 . . « denote quite distinct coefficients, % being the
dependent variable in the first case and in the second.

A coefficient with p secondary subscripts may be termed a regression
of the -pth order. The regressions ^ 2 i> ^is» obtained by con-

sidering two variables alone, may be regarded as of order zero, and may
be termed total, as distinct from partial, regressions.

12.6 If the regressions ^ 13 . 24 .. n> be assigned the

" best '' values, as determined by the method of least squares, the difference
between the actual value of and the value assigned by the right-hand
side of the regression equation (12.1), that is, the error of estimate, will be
denoted hy 23 . . . n J Q-s a definition we have —

%.28 . . . ^12,84 . . . n^2 ^13.24 . . . n^3 • • * ^in,28 . . . (12,2)

where ^ 2 , • . . sire assigned any one set of observed values. Such an
error (or residual, as it is sometimes called), denoted by a symbol with p
secondary sufiixes, will be termed a deviation of the ^th order.

Finally, we will define a generalised standard deviation crj, 23 . . . «
the equation

. . . (12.3)

IV being, as usual, the number of observations. A standard deviation
denoted by a S 5 mibol with p secondary suffixes* will be termed a standard

PARTIAL CORRELATION

285

deviation of the ;^th order, the standard deviations Og, etc., being
regarded as of order zero, the standard deviations ( 7 ^ 2 , first

order, and so on,

12.7 In the case of two variables, the correlation coefficient
be regarded as defined by the equation

We .shall generalise this equation in the form

^ 12.34 . . n = (^12.34 . . n^21.34 . . • * • (12.4)

This is at present a pure definition of a new symbol, and it remains to be
shown that ^ 12.34 ... « niay really be regarded as, and possesses all the pro-
perties of, a correlation coefficient ; the name may, however, be applied
to it, pending the proof. A correlation coefficient with p secondary
subscripts will be termed a correlation of order p. Evidently, in the
case of a correlation coefficient, the order in which both primary and
secondary subscripts is written is indifferent, for the right-hand side of
equation (12.4) is unaltered by writing 2 for 1 and 1 for 2. The correla-
tions ^i 3 > ^^ly be regarded as of order zero, and spoken of as total,
as distinct from partial, correlations.

The normal equations

12.8 AH the quantities we have just defined are expressible in terms
of the total and partial regression coefficients, and particular importance
therefore attaches to the equations which give those coefficients. The
equations of 12.4 may be written

^(^2%.23 . . .... (12,5)

etc., there being (^— 1 ) equations for each regression equation.

These equations are called the normal equations,

12.9 If the student will follow the process by which (12.5) was obtained,
he will see that when the condition is expressed that & 12 . 34 . . . n shall
possess the ieast*square value, enters into the product-sum with
^ 1.23 ... n I when the same condition is expressed for ^ 13,24 ... n> ^3 enters
into the product-sum, and so on. Taking each regression in turn, in fact,
every % the suffix of which is included in the secondary suffixes of % 33 ^ ^ «
enters into the product-sum. The normal equations of the form (12.5) are

^therefore equivalent to the theorem —

The product-sum of any deviation of order zero with any deviation of higher
order is zero, provided the subscript of the former occur among the secondary
subscripts of the latter.

286

THEORY OF STATISTICS

12.10 But it follows from this that

^(^1 34 . . 34 . . . 34 . . n(^2-^23 4 . . n^3-* • •-^2n 34 .

Similarly,
Similarly again

.«^ 2 )

^(■*1 31

. n^2 34

n)”^ (^1^2 34 n)

n^2 34 . .

. (n-l))=^^(^l 34 . n^2j

and so on. Therefore, quite generally,

^(^1 34 n^2 34 . . . 34 • • • (n-l)^'2 34 n)

. (n~l)^n) }

— 34 n) 1

34 . n^2 34 (n-l)) |

(12.6)

^(^3.34 . n^a) /

Comparing all the equal product-sums that may be obtained in this way,
we see that the product-sum of any two deviations in which all the secondary
subscripts of the first occur among the secondary subscripts of the second is
unaltered by omitting any or all of the secondary subscripts of the first, and,
conversely, the product-sum of any deviation of order p with a deviation of
order the p subscripts being the same in each case, is unaltered by adding
to the secondary subscripts oj the former any or all of the q additional sub-
scripts of the latter,

It follows therefore from (12.$) that any product-sum is zero if all the
subscripts of the one deviation occur among the secondary subscripts of the
other. As the simplest case, we n>ay note that is uncorrelated with x,^
and X 2 uncorrelated with % 2 -

The theorems of this and of the preceding paragraph are of fundamental
importance, and should be carefully remembered.

12.11 We can now show that the quantities r defined by (12.4) are
really coefficients of correlation. In fact we have, from the results of
12.9 and 12.10,

PARTIAL CORREtATlOl^

287

and determining 34 „by the method of least squares, i.e. 34 ^

is the regression of _ n on ^2 34 . n- It follows at once from

(12.4) that ri2 34 . . n is the correlation between 34 „ and a ;2 34 . . .

and from (12.7) that we may write

^12 34 . . . n=^l2 34 . . ^ • • • (12.8)

^2 34 . . n

an equation identical with the familiar relation ^i2=^20‘i /og, with the
secondary suffixes 34 ^ added throughout.

To illustrate the meaning of the equation by the simplest case, if we had
three variables only, and x^, the value of 3 or 3 could be

determined (1) by finding the correlations and and the corresponding
regressions 643 and b^^ ; (2) working out the residuals —b^^x^ and x^—
^23^3 lor all associated deviations ; (3) working out the correlation

between the residuals associated with the same values of The method
would not, however, be a practical one, as the arithmetic would be extremely
lengthy, much more lengthy than the method given below for expressing
a correlation of order p in terms of correlations of order p—l.

Expression of standard deviation in terms of standard deviations and
coefficients of lower orders

12.12 Any standard deviation of order p may he expressed in terms of a
standard deviation of order p— 1 and a correlation of order p— 1. For,

.23 . . n)^— ^(^1 23 . . . (n~l)^i.23 . . n)

==2 :(a ;,.23 . . 23 {n-i):^n -terms in a ;2 to

=S(A;f 23 . . (n-l)) ^in 23 . . (n-l)^(^l 23 (n-l)^n 23 (n-l))

or, dividing through by the number of observations —

^1-23 • * * n~^1.23 . . . {n-‘l)(I ^lr» 23 . . . (n-l)^nl 23 . . (n-l))

~^1.23 . . . (»— l)(I ^1«.23 . . (n-l)) • • • (12.9)

This is again the relation of the familiar form

with the secondary suffices 23 . . . ^n~l) added throughout. It is clear
from (12.9) that 23 . . . like any correlation of order zero, cannot be
numerically greater than unity. It also follows at once that if we have
been estimating x^ from a? 3, . . . will not increase the accuracy

of estimate unless 2 23 . . . («-i) ^m) differ from zero. This condition

is somewhat interesting, as it leads to rather unexpected results. For
example, if fi2=+0-8, ri3=+0*4, r23=+0*5, it will not be possible to
estimate % with any greater accuracy from x^, and than from alone,
for the value of is zero (see below, 12.15).

2§8

THEORY OF STATISTIC^

12,13 It should be noted that, in equation (12.9), any other subscript
can be eliminated in the same way as subscript n from the suffix of
23 n> so that a standard deviation of order p can be expressed in p
ways in terms of standard deviations of the next lower order. This is useful
as affording an independent check on arithmetic. Further, cTj 23 . . . (n-i)
can be expressed in the same way in terms of 23 (n~ 2 )> so on, so

that we must have

23 . . .n— 14 23) * * • (l’“^ln 2 S . (n-l)) (12.10)

This is an extremely convenient expression for arithmetical use ; the
arithmetic can again be subjected to an absolute check by eliminating the
subscripts in a different, say the inverse, order. Apart from the algebraic
proof, it is obvious that the values must be identical ; for if we are
estimating one variable from n others, it is clearly indifferent in what
order the latter are taken into account.

^ 1 . 23 ...» can also be expressed in terms of and the total correlation
coef&cients. We have

^(%.23 . . . n)^“^{%(%.23 •■*«)} =^^1.23 . . . n

Hence, expanding % 23 . n»

^1 ^12 3 . . ^13 2 . . . n^l3^1*^8"“ * • • 23 * * • n

The normal equations involving %. 28 . n ^.re

2(a? 2%.28 . , . n)“^# etc.

i.e. expanding,

^ 21 ^ 1^2 ^12 3 . . . n^t ^13 2 . . . 22 ,^ 2^2 • • •

— ^12 3 . . . v^Z2^Z^2 ^13 2 . . . nPz • • * CtC.

Regarding the n equations so obtained as equations in the quantities b,
we have, on elimination, the determinant

23 . . n r^2^1<^2 ^13^10-3 — - ^InCTiCTn

21^ 2^1 ^2 ^23^2^2 • • • ^2n^2^n q

Dividing the sth row by and the tth column by a^, this gives —

^12

^13 • •

• rin

^21

^28 • •

■ f2n

^nx

^«2

rns • •

. 1

PARTIAL CORRELATION

289

Write (o for the determinant

I 1

ri2 . . .
1 ...

and let be the minor of the term in the first row and column. Then

^123 n_ A
0 ) 0 ^ 11=0

•'1
0-1 23

Similarly,

(i>

. ( 12 . 11 )

^2 IS

C£>,

and so on.

These results exhibit oj 23 n» ^ symmetrical form.

Expression of regression coefficients in terms of coefficients of lower orders

12.14 Any regression of order p may be expressed in terms of regressions
of order p—l. For we have —

1.84 . . n^2.34 . . fi) — 34 (n-l)^2 34 , . n)

=S(% ^4 . . (n-i)){^2-'^2n.34 . . {n-D^n “terms in a ;3 to

==S (^1 34 ^ (n-l)^2.34 . . (n-l)) ^2n 34 . . {n-l)^(^1.34 . . in-i)^n 34 . . (n— l))

Replacing 34 . („_!) by 34 . . . . (n~i)/^2 34 . . (n-i)

we have —

i.84 . . n^2.34 . n^^l2 34 . {n-l)‘^I.S4 . (n-l) ^In 34 . . (n-l)^«2.34 . . {n-l)*^|.84 . (n-l)

or, from (12.9),

^^12 34

_ ^12 34 . (n~l) — ^in 84 (n-l)^w2 34 , (n-l)

1 b^n 34 . . (n~l)^n2.34 . . (n-l)

The student should note that this is an expression of the form

(12.12)

I, __^12 \rfinZ

^12.n'~TZjr7i

1 — 02nPm

with the subscripts 34 . . . («— 1) added throughout. The coefficient
^ 12.34 . . « therefore be regarded as determined from a regression
equation of the form

^1.34 . . . (n-l) “^12.34 . . . n^2.34 . . , (n-l) “b^ln.aS . . {n-l)^n.34 . . . (w-l)

i.e. it is the partial regression of % 34 , _ («„ij on 34 . , 34 , , ,

being given. As any other secondary suffix might have been eliminated
in lieu of », we might also regard it as the partial regression of ^ ^ , u
on Xj 45 _ , ^^3 45 ^ being given, and so on.

290

THEORY OE STATISTICS

Expression of correlation coefficient in terms of coefficients of lower
orders

12.15 From equation (12.12) we may readily obtain a corresponding
equation for correlations. For (12.12) may be written —

h ^12 34 (n~l) ^in 34 (n~l)^2n 34 (n-l) ^134 (n-l)

^12.34 n— rZ~72 ; ^

^ ^2n 34 . . (n-l) ^2 34 . . . (n-l)

Hence, writing down the corresponding expression for 621.34 . . n
taking the square root —

^12 34 (n-l) ^in 34 _ (n-l)^2n 34 (n-l)

^in 34 (n-l))^(l ^2n 34 (n~l)^

This is, similarly, the expression for three variables—

(12.13)

^ ^12~^ln^2n

with the secondary subscripts added throughout, and 34 . n can be
assigned interpretations corresponding to those of 6^2 34 . . n above.
Evidently equation (12.13) permits of an absolute check on the arithmetic
in the calculation of all partial coefficients of an order higher than the
first, for any one of the secondary suffixes of 34 n can be eliminated
so as to obtain another equation of the same form as (12.13), and the
value obtained for ^12.34 . . n t>y inserting the values of the coefficients
of lower order in the expression on the right must be the same in each case.

Practical procedure

12,16 The equations now obtained provide all that is necessary for
the arithmetical solution of problems in multiple correlation. The best
mode of procedure on the whole, having calculated all the correlations
and standard deviations of order zero, is (1) to calculate the correlations
of higher order by successive applications of equation (12.13) ; (2) to
calculate any required standard deviations by equation (12.10) ; (3) to
calculate any required regressions by equation (12.8) ; the use of equation
(12.12) for calculating the regressions of successive orders directly from
one another is comparatively clumsy. We will give two illustrations,
the first for three and the second for four variables. The introduction of
more variables does not involve any difference in the form of the arithmetic,
but rapidly increases the amount.

Example 12.1. — In Exercise 9.2, page 234, we gave some data of (1)
the average earnings of agricultural labourers, (2) the percentage of the
population in receipt of poor law relief, (3) the ratios of the numbers in
receipt of outdoor relief to those relieved in the workhouse, for 38 rural
districts. Required to work out the partial correlations, regressions, etc.,
for these three variables.

PARTIAL CORRELATION

291

Using as our notation A''i=average earnings, A2=:percentage of
population in receipt of relief, Z3=out-relief ratio, the first constants
determined are —

Mi~15*9 shillings shillings ri2~--0-66

^2= 3*67 per cent Og— 1 -29 per cent ri3=— 0*13

^3= 5-79 03=3-09 r23=+0-60

To obtain the partial correlations, equation (12.13) is used direct in
its simplest form —

y _ ^12 ^13^23

The work is best done systematically and the results collected in
tabular form, especially if logarithms are used, as many of the logarithms
occur repeatedly. First, it will be noted that the logarithms of (1 —
occur in all the denominators ; these had, accordingly, better be worked
out at once and tabulated (col. 2 of the table below). In column 3 the
product term of the numerator of each partial coefficient is entered, i.e.

logV l — r*

Product

term

Numera-

tor

log

num.

log

denom.

7 8

Correlation of
first order

Iog\/ 1-f’

log

Value

f„= — 0 66

fis = ~0 13

f8,=“ + 0 60

1 87580

1- 99629

1 90309

^ -0 0780
-0 3960
+ 0-0858

-0 5820
+0 2660
+0 5142

I 76492

1 42488
1-71113

I 89938

I 77889
1-87209

I 86554
T 64599
1-83904

f»,-0 73

ns s+o 44

ns 1+0 69

I 83216

I 95267

I 85946

the product of the two other coefficients on the remaining lines in column 1 ;
subtracting this from the coefficient on the same line in column 1, we have
the numerator (col. 4) and can enter its logarithm. The logarithm of the
denominator (col. 6) is obtained at once by adding the two logarithms of
(1 — on the remaining lines of the table, and subtracting the logarithms
of the denominators from those of the numerators, we have the logarithms
of the correlations of the first order. It is also as well to calculate at
once, for reference in the calculation of standard deviations of the second
order, the values of log ■— H for the first-order coefficients (col. 9).

Having obtained the correlations, we can now proceed to the regressions.
If we wish to find all the regression equations, we shall have six regressions
to calculate from equations of the form

^12 3 ^^12 3^ 1.3 3

These will involve all the six standard deviations of the first order

^ 2 . 1 * ^2.3» "The standard deviations of the first order are not

tHEORV OF StATlSTiCS

i29:2

in themselves of much interest, but the standard deviations of the second
order are important, as being the standard errors or root-mean-square errors
of estimate made in using the regression equations of the second order.
We may save needless arithmetic, therefore, by replacing the standard
deviations of the first order by those of the second, omitting the former
entirely, and transforming the above equation for &12.3 the form

^ 12 . 3 ^^ 12 . 8 ^ 1.23 /^ 2.13

This transformation is a useful one and should be noted by the student.
The values of each a may be calculated twice independently by the formulas
of the form

=CTi{l-rf3)i(l -rfa

so as to check the arithmetic ; the work is rapidly done if the values of
log V 1 have been tabulated. The values found are —

log Oj 23=0*06146 O’! 23=1*15

log cr2.i3=T* 84584 Og i3=0*70

log cjg 12=0-34571 ag 12=2*22

From these and the logarithms of the r's we have —

log *’12.3=0*08116,

^12 3~ — f ‘21

log ^^13 2=1*36174

^13 2

= 4-0-23

^og 62 i.3=T"64993,

^21. 3~ — 0*45

log 623 1=1.33917

^23.1

= 4-0-22

*og 631. a =1-93024,

^31. 2~ +0*85

log 632.1=0.33891

^32 1

= 4-2-18

That is, the regression equations are —

(1) %=— 1* 21^2+0 *23^t;3

(2) = —0* 45% +0* 22:^3

(3) ^ 3 = — j”0 * 85% ~j~2 • 18 j3c^2

or, transferring the origins to zero —

(1) Earnings Xi=-|-19*0— I *21^2+0*23X3

(2) Pauperism Xg = +9 * 55 -0 * 45Xi +0 • 22X3

(3) Out-relief ratio X3=~-15*7+0*85Xi+2* ISXg

The units are throughout one shilling for the earnings Xi, 1 per cent for the
pauperism Xg and 1 for the out-relief ratio X3.

Now let us examine the light thrown by these results on the relationship
between the variables.

The first and second regression equations are those of most practical
importance. The argument was once advanced that the giving of out-
relief tended to lower earnings, and the total coefficient (%3=—0-13)
between earnings (Xi) and ont-relief (Xg), though very small, does not
seem inconsistent with such a hypothesis. The partial correlation
coefficient (fi3.2j=+0*44) and the regression equation (1), however.

PARTIAL CORRELATION

293

indicate that in unions with a given percentage of the population in receipt
of relief {X^ the earnings were highest where the proportion of out-relief
was highest ; and this is, in so far, against the h 3 ?pothesis of a tendency
to lower wages. It remained possible, of course, that out-relief might
adversely affect the possibility of earning, e.g. by limiting the emplo 5 mient
of the old.

As regards pauperism, the argument might be advanced that the
observed correlation (^ 23 = 4 - 0 * 60 ) between pauperism and out-relief was
in part due to the negative correlation — 0 * 13) between earnings and

out-relief. Such a hypothesis would have little to support it in view of the
smallness and doubtful significance of and is definitely contradicted
by the positive partial correlation ^ 23.1 = +^^*^^ and the second regression
equation. The third regression equation shows that the proportion of
out-relief was on the whole highest where earnings were highest and
pauperism greatest. It should be noticed, however, that a negative ratio
is clearly impossible, and consequently the relation cannot be strictly
linear ; but the third equation gives possible (positive) average ratios for
all the combinations of pauperism and earnings that actually occur.

Example 12.2 {Four variables ), — ^As an illustration of the form of the
work in the case of four variables, we will take a portion of the data from
another investigation into the causation of pauperism.

The variables are the ratios of the values in 1891 to the values in 1881
(taken as 100 ) of —

1 . The percentage of the population in receipt of relief,

2. The ratio of the numbers given outdoor relief to the numbers relieved

in the workhouse,

3. The percentage of the population over 65 years of age,

4. The population itself,

in the metropolitan group of 32 unions, and the fundamental constants
(means, standard deviations and correlations) are as follows —

TABLE 12.1

Means

Standard

deviations

Correlation

coefficient

log \/l— r*

104-7

29-2

-fO-52

1-93154

90-6

41-7

+0-41

1-96003

107-7

5-5

-0-14

1-99570

111-3

23-8

+0-49

T- 94038

-fO-23

1-98820

—

— «

—

+0-25

1-98598

294

THEORY OF STATISTICS

It is seen that the average changes are not great ; the percentages of the
population in receipt of lelief increased on an average by 4*7 per cent,
the out-rehef ratio dropped by 9*4 per cent and the percentage of the
old increased by 7*7 per cent, while the population of the unions rose
on the average by 11-3 per cent. At the same time the standard devia-
tions of the first, second and fourth variables are very large. As a matter
of fact, while in one union the pauperism decreased by nearly 50 per cent
and in others by 20 per cent, m some there were increases of 60, 80 and

TABLE 12.2

Correlation
coefficient
(zero order)

Product
term of
numerator

Numerator

Correlation
coefficient
(first order)

log 1 —r^

+ 0-52

+ 0*2009

+0 3191

12*3

+0*4013

T* 96187

+0-41

+ 0*2548

+0*1552

13*2

+ 0*2084

1*99035

+0 49

+ 0*2132

+ 0*2768

23*1

+0*3553

1*97070

■f 0 52

-0*0322

+0*5522

12-4

+0*5731

1*91355

-0-14

+ 0*1196

-0*2596

14*2

-0*3123

1*97772

4-0*23

-0 0728

+ 0*3028

24*1

+0*3580

1 97022

+0 41

-0*0350

+ 0*4450

13*4

+0*4642

1*94731

-0 14

+0*1025

-0*2425

14*3

-0*2746

1 98297

4*0 • 25

-0*0574

+0*3074

34*1

+ 0*3404

1*97326

4-0*49

+ 0*0575

+0*4325

23*4

+0*4590

1*94863

+0*23

+0*1225

+0*1075

24*3

+0*1274

T* 99645

+ 0*25

+ 0*1127

+ 0*1373

34*2

+ 0*1618

1 *99424

90 per cent ; similarly, in the case of the out-relief, in several unions the
ratio was decreased by 40 to 60 per cent, a consistent anti-out-relief
policy having been enforced ; in others the ratio was doubled, and more
than doubled. As regards population, the more central districts showed
decreases ranging up to 20 and 25 per cent, the circumferential districts
increases of 45 to 80 per cent. The correlations of order zero are not
large, the changes in the rate of pauperism exhibiting the highest correlation
with changes in the out-relief ratio, slightly less with changes in the
proportion of old and very little with changes in population.

The correlations of the second order are obtained in two steps. In the
first place, the six coefficients of order zero are grouped in four sets of three,
corresponding to the four sets of three variables formed by omitting each
one of the four variables in turn (Table 12.2, col. 1). Each of these sets
of three coefficients is then treated in the same manner as in the last
example, and so the correlations of the first order (Table 12.2, col. 4) are
obtained. The first-order coefficients are then regrouped in sets of three,
with the same secondary suffix (Table 12.3, col. 1), and these are treated
precisely in the same way as the coefficients of order zero. In this way, it

PARTIAL CORRELATION

295

will be seen, the value of each coefficient of the second order is arrived at in
two ways independently, and so the arithmetic is checked : 34 occurs in

the first and fourth Hues, for instance, ^13 34 in the second and seventh, and
so on. Of course slight differences may occur in the last digit if a sufficient
number of digits is not retained, and for this reason the intermediate work
should be carried to a greater degree of accuracy than is necessary in the
final result ; thus four places of decimals were retained throughout in the
intermediate work of this example, and three in the final result. If he
carries out an independent calculation, the student may differ slightly
from the logarithms given in this and the following work, if more or fewer
figures are retained.

TABLE 12.3

Correlation
coefficient
(first order)

Product
term of
numerator

Numerator

Correlation
coefficient
(second order)

log V^l— f*

12-4

+ 0 5731

+0-2131

+0 3600

12-34

+ 0-457

1*94901

13*4

+ 0 4642

+0-2631

+0-2011

13-24

+0-276

1-98277

23-4

+ 0-4590

+ 0-2660

+0-1930

23-14

+ 0-266

1-98408

12-3

+ 0-4013

-0-0350

+0-4363

12-34

+0-457

—

14*3

-0-2746

+ 0-0511

-0-3257

14-23

-0-359

T- 97013

24*3

+0-1274

-0-1102

+0-2376

24-13

+0-270

T- 98359

13-2

+ 0-2084

-0 0505

+ 0-2589

13-24

+ 0-276

14-2

-0-3123

+0-0337

-0.3460

14-23

-0-359

—

34-2

+0-1618

-0-0651

+0*2269

34-12

+0-244

1-98664

23*1

+0-3553 1

+ 0 1219

+0-2334

23-14

+0-266

—

24-1

+ 0-3580

+ 0-1209

+0-2371

1 24*13

+0*270

—

34*1

+0-3404

+ 0*1272

+0-2132

1 34-12

+0-244

Having obtained the correlations, the regressions can be calculated from
the third-order standard deviations by equations of the form (as in the last
example),

h 234

^12.34 U-Z

^2 134

so the standard deviations of lower orders need not be evaluated,
equations of the form

a'l.234=<^l(l -^13 2)*(1 -»'l4 23)^

we find :

log o'j 234 — 1 -35740

log <^2 134=1-50597

log 0^3 ,24 ”11-05773

^3-124~ 4*55

log <^4.128=1 '32914

Using

296

tHEORY OF STATISTICS

All the twelve regressions of the second order can be readily calculated,
given these standard deviations and the con*elations, but we may confine
ourselves to the equation giving the changes in pauperism (Xi) in terms of
other variables as the most important. It will be found to be

=0 • 325^2 + 1 —O • 383a;4

or, transferring the origins and expressing the equation in terms of per-
centage ratios,

Xi = -3M +0 • 325X2 + 1 • 383X3 -0 • 888X4

or, again, in terms of percentage changes (ratio — 100) —

Percentage change in pauperism

= +l‘4 per cent

+0*325 times the change in out-relief ratio
+1*383 „ „ „ proportion of old

—0*383 „ „ „ population

These results render the interpretation of the total coefficients, which
might be equally consistent with several hypotheses, more clear and definite.
The questions would arise, for instance, whether the correlation of changes
in pauperism with changes in out-relief might not be due to correlation of
the latter with the other factors introduced, and whether the negative
correlation with changes in population might not be due solely to the
correlation of the latter with changes in the proportion of old. As a matter
of fact, the partial correlations of changes in pauperism with changes in
out-reUef and in proportion of old are slightly less than the total correla-
tions, but the partial correlation with changes in population is numerically
greater, the figures being —

^ 12 . 34 “

?'i3 24=+0‘28

^14“ 0*14 ^14 23~ 0*36

So far, then, as we have taken the factors of the case into account, there
appears to have been a true correlation between changes in pauperism and
changes in out-relief, proportion of old and population — ^the latter serving,
of course, as some index to changes in general prosperity. The relative
influences of the three factors are indicated by the regression equation
above.

In this and the previous example we have had to consider only three
or four independent variables. For five or more the number of partial
correlations and regressions increases rapidly (see Exercise 12.6) and it
becomes impracticable to compute them aU without great labour. In such
circumstances, where we are primarily interested in the regression of one
vanate on the others it may well be easier to solve direct the normal
equations given at the end of 12.4, either by progressive elimination of

PARTIAL CORRELATION 297

variables in the usual manner for simultaneous linear equations or by
evaluating determinants systematically. See the comments on this point
in 13.27-13.29.

Aids to calculation

12,17 lo facilitate the computation of ,artial correlation and regression
coefficients, various tables of such qua.^i.:t:es as

have

VI— r^,

^‘'^cpared. See, for iuhtance, T. L. Kelley's SlaU‘sh‘cal Tables,

The generalised scatter diagram

12.18 The scatter diagram in two dimensions may be generalised to
three dimensions, and may also be used as a mental construct for higher
dimensions, though no actual model can of course be made.

Consider the case of three variates. The values of and

associated with any given individual may be regarded as determining a
point in space whose co-ordinates are and X3. The totality of

individuals will therefore give us a swarm of points in three-dimensional
space, which will lie distributed in certain ways about planes of regression.
The closeness with which the points lie to the regression planes is a
measure of the adequacy of the representation by regression equations.
In figure 12.1 we give a diagrammatic representation of the data of
Example 12.1 with the regression plane of on the other two variables.

Fig. 12.1. — Generalised scatter diagram for three variables
Data of Example 12.1. average earnings, A'a^percentage of population in
receipt of relief. As® out-relief ratio.

298

THEORY OF STATISTICS

Coefficient of multiple correlation

12.19 Consider the regression equation for

Xi = bji^2 3 71^3 2 . 2 . (n~l)^n

Let us write the right-hand side of this equation as 23 . . . n» so that in
virtue of (12.2),

^1,23 . n~-^l 23 n • • . (12.14)

Now consider the correlation between x^ and 23 . n- We have

in virtue of the theorem of 12.10 —

^(^1^123 n)— ^123 n)}

=S(a:i^) ^{Xi{Xj 23 . . n)}

..n)

Also,

2(^1 23 . . % 23 . . . n)^

=iV(a2-CT2

Hence, the correlation between x^ and 23 . n

gi-o-iaa ■ n
CTiVcrf-aJ 23 „

■ ..n

We shall call this quantity „). We have immediately —

^1 23 . . -^1(23 . . . n)) • • * (12.15)

•^1(2 * • • n) is called the multiple correlation coefficient between % and
We have, similarly, multiple correlations between % and
fewer variables. Rj ^2 «) is called an (fj— l)-fold multiple correlation
coefficient. R ^{2 . . ;j=l) would be an (^— 2)-fold coefficient, and so on.

12.20 The value of R may be calculated either directly from equation
(12.15), or by substituting in that equation the value of aj 23 n obtained
in (12.10), which gives —

1 i?|(23 . . . ^I2)(i““^13 2)(i ”^14.23) • * ‘ (i ^l.n23 . . . (n-l)) (12.16)

Properties of the multiple correlation coefficient

12.21 2fi(23 . . n)^ being the correlation between x^ and 23 . . m

measures how closely can be represented by the regression equation. If
R^l, % can be perfectly represented by such an equation, i.e. is a linear
function of ^2 . . . In this case of 23 _ „=0, i.e, all the residuals are

zero.

PARTIAL CORRELATION

299

It may, in fact, be shown that „> is greater than the correlation

between and any linear function of x^.. .x„ other than that expressed
in the regression equation, i.e. 23 . , Putting this another way, the
regression coefficients in „ may be determined by the condition

that the correlation between iCj and fii.as , . . « is a maximum.

R is necessarily positive or zero

12.22 This is true, since the product term 23 „) is positive,

being equal to iV(a|— 0*33 „), and we see from (12.10) that a|>

23 n*

Further, from (12.16),

1 ^1(23

i.e. R is not numerically less than Similarly, it is not numerically less
than any other total or partial correlation coefficient which can appear
in (12.16). Hence, jRi (2 ts not numerically less than any possible
constituent coefficient of correlation.

It follows from this that if ^ ^ n)=0, ail the correlation coefficients
involving are zero, i.e. the variate % is completely uncorrelated imth the
other variates.

12.23 Further, even if ail the variables Xi, Xg, . . . X„ were strictly
uncorreiated in the original population as a whole, we should expect r^^*
Hz 2 » ^14 23 * to exhibit values (whether positive or negative) differing
from zero in a limited sample. Hence, R will not tend, on an average
of such samples, to be zero, but will fluctuate round some mean value.
This mean value will be the greater the smaller the number of observations
in the sample, and also the greater the number of variables. When only
a small number of observations is available it is, accordingly, little use to
deal with a large number of variables. As a limiting case, it is evident
that if we deal with n variables and possess only n observations, all the
partial correlations of the highest possible order will be unity. We shall
deal with the question of the significance of an observed vffiue of R in
Chapter 22.

Example 12.3. — In Example 12.1 we found—

#"12=— 0-66

Hence, from (12.16),

l-i?|(28)-{l-(0-66)2}{l-(0-44)n

=0-455

whence

Ri(2s)=0-74

300

THEORY OF STATISTICS

Similarly, it will be found that

■^2(13)

and

■^^ 3 ( 12 ) =0*70

The student may verify by inspection that these values are greater than
the corresponding constituent values.

Expression of regressions and correlations in terms of coefficients of
higher orders

12.24 It is obvious that as equations (12.12) and (12.13) enable us to
express regressions and coi relations of higher orders in terms of those of
lower orders, we must similarly be able to express the coefficients of lower
in terms of those of higher orders. Such expressions are sometimes useful
for theoretical work. Using the same method of expansion as in previous
cases, we have —

That is,

0=2(^i 23 34 (n-l))

‘”^(•^1^2 a4 . (n-i)) ^12 34 . . 34 . (n-l))

^in 23 . 34 . . . (n-l))

^12 34 . (n-l)— ^12 34 n+^ln 23 . (n-l)^n2 34 . . (n-l)

In this equation the coefficient on the left and the last on the right are of
order M— 3, the other two of order n— 2. We therefore wish to eliminate the
last coefficient on the right. Interchanging the suffixes 1 for n and.n for
1, we have —

34 . . (n-l)— ^n2.13 . (n-l) "i'^nl.23 . (n-l)^12 34 , . (n-l)

Substituting this value for 34 . , (n-i) in the first equation, we have-

^12 34 . . (n-l) =

_^12 34 n~b^ln 23 * • ■ (n-l)^n2 13 . (n-l)

I ^In 23 . . (w-l)^nl.23 . (n-l)

This is the required equation for the regressions ; it is the equation

h ^12.n~t“^ln 2^n2 1

^12— I

* ^in 2^nl.2

(12.17)

with secondary suffixes 34 . . . (w—1) added throughout. The corre-
sponding equation for the correlations is obtained at once by writing down

PARTIAL CORRELATION

301

equation (12.17) for 34 and taking the square root of the

product , this gives —

^'12 34 n 23 in-l)^2n 1 3 (n-l )

^3 n 23 . (n--l))"(i 13 (rj-l))^

(12.18)

vvliich IS similarly the equation

y — - 2 ^ 2 ri x

with the secondary suffixes 34 ... (;^ -1) added throughout.

Conditions of consistence among correlation coefficients
12.25 Equations (12.13) and (12.18) imply that certain limiting inequali-
ties must hold between the correlation coefficients in the expression on
the right in each case in order that real values (values between if)
be obtained for the correlation coefficient on the left. These inequalities
correspond precisely with those '' conditions of consistence between
class-frequencies with which we dealt in Chapter 1, but we propose to treat
them only briefly here. Writing (12.13) in its simplest form for we
must have rf 2 3 <1 or

n-rh){l-rh)'-'

that is,

''12 +''?3 +>'|3-2>'i2>'i3»'S3

. (12.19)

if the three r’s are consistent with one another. If we take >'j3 as
known, this gives as limits for ^23*

rnh3±\/l -rf^-rls-irf^rl3

Similarly, writing (12.18) in its simplest form for in terms of 3>
fi3 2 and ^23 1> we must have —

^12 3^13 2^23 * * (12.20)

and therefore, if ^43.3 ^.nd 2 are given, 1 niust lie between the limits

””■^12 3^13 2i Vi ^'12 3 ^13 2"h^i2 3^13 2

The following table gives the limits of the third coefficient, in a few

302 THEORY OF STATISTICS

special cases, for the three coefficients of zero order and of the first order
respectively —

Value of

Limits of

^12 8

^13 ^13.2

^23

^23 1

+ 1

±1

+ 1

-1

±V0-5

iVo-s

0, il

0, -I

+ ^0-5

TVO-5

0, -I

0,ii

The student should notice that the set of three coefficients of order zero
and value unity are only consistent if either one only, or ail three, are
positive, i.e. +1, +1, +l>or —1, —1, +1 ; but not ~1, —1, ~L On the
other hand, the set of three coefficients of the first order and value unity
are only consistent if one only, or all three, are negative : the only con-
sistent sets are +1, +1, —1 and —1, —1, -1. The values of the two
given r’s need to be very high if even the sign of the third can be inferred ;
if the two are equal, they must be at least equal to VO-5 or 0-707 . , ,
Finally, it may be noted that no two values for the known coefficients ever
permit an inference of the value zero for the third ; the fact that 1 and 2,
1 and 3 are uncorrelated, pair and pair, permits no inference of any kind
as to the correlation between 2 and 3, which may lie anvAvhere between
+1 and —1.

Fallacies in the interpretation of correlation coefficients
12.26 We do not think it necessary to add to this chapter a detailed
discussion of the nature of fallacies on which the theory of multiple correla-
tion throws much light. The general nature of such fallacies is the same
as for the case of attributes, and was discussed fully in Chapter 2. It
suffices to point out the principal sources of fallacy which are suggested
at once by the form of the partial correlation

(a)

and from the form of the corresponding expression for in terms of the
partial coefficients —

From the form of the numerator of (a) it is evident (1) that even if be
zero, ^ 12,3 wiJl ^ot be zero unless either or or both, are zero. If
and f 23 are of the same sign, the partial correlation will be negative ; if of

PARTIAL CORRELATION

303

opposite sign, positive. Thus the quantity of a crop might appear to be
unaffected, say, by the amount of rainfall during some period preceding
harvest : this might be due merely to a correlation between ram and
low temperature, the partial correlation between crop and rainfall being
positive and important. We may thus easily misinterpret a coefficient of
correlation which is zero. (2) 3 may be, indeed often is, of opposite

sign to ^12, and this may lead to still more serious errors of interpretation.

From the form of the numerator of (b), on the other hand, we see that,
conversei}^ will not be zero even though ?'i2 3 is zero, unless either
^13.2 ^23 1 is zero. This corresponds to the theorem of 2.26, and indicates

a source of fallacies similar to those there discussed.

12.27 We have seen that 3 is the correlation between 3 and 3, and
that we might determine the value of this partial correlation by drawing
up the actual correlation table for the two residuals in question. Suppose,
however, that instead of drawing up a single table we drew up a series of
tables for values of x^ 3 and ^^2^3 associated with values of lying within
successive class-intervals of its range. In general, the value of 3 would
not be the same (or approximately the samel for all such tables, but would
exhibit some systematic change as the value of increased. Hence
should be regarded, in general, as of the nature of an average correlation :
the cases in which it measures the correlation betv/een and ^^2.3 for
every value of (cf. below 12 . 31 ) are probably exceptional. The process
for determining partial associations (cf. Chapter 2) is, it wiU be remembered,
thorough and complete, as we always obtain the actual tables exhibiting
the association between, say, A and B in the population of Cs and the
population of y's: that two such associations may differ materially is
illustrated by Example 2.9, page 34. It might sometimes serve as a useful
check on partial correlation work to reclassify the observations by the
fundamental methods of Chapter 2.

Multivariate normal correlation

12.28 The theorems and results of Chapter 10 in regard to notmal
correlation can be extended to the case of n variates, which we have studied
in this chapter.

In fact, suppose we have n variates x^ ^3, . . . measured from
their respective means, with standard deviations cr^ Og, . . . Let
us first consider the simple case in which they are normally distributed
and each is completely independent of the others.

Then, if . . . „ denote the frequency of the combination of deviations
Xi, X2, • * ^ Xn, we have —

where

( 12 . 21 )

304

THEORY OF STATISTICS

Now consider the variates X 21 , X 2 22 f ^ 12 . . (n-D* Whether

x^, X 2 t • • . Xn are correlated or not, these variates are uncorrelated, in
virtue of 12.10. Let us further suppose they are independent and normally
distributed. Then their distribution is given by

where

and

JV12 • * • ^^2 1 iz • (n- i))

(n-l)j— 2+^2 ^ *

1 2 (w-l)

0'».i2 . (n-l)

(277)2cJia2i. . . 0^12

( 12 . 22 )

( 12 . 23 )

( 12 . 24 )

The expression (12.23) may be put in a more convenient form. It may
be shown, but we omit the proof, that

-'l 23

-'2 13

(n-l)

x.x

1-^2

n‘^2 13 ... n

^^in-l)n 12 (n-2)

it'n

0“n-3 .1

{n~2)n^n 1

(n-l)

( 12 . 25 )

which exhibits the form as symmetrical in %
Now we showed in 12.13 that

23 . n~

etc.

In precisely the same way it may be shown that

I 0)

Oi.28 . . . n^2 13 «ri2 3 . n — “■7~<^l’^2

^12

0)^2 being the minor in o) of the term in the first row and the second
column.

If we substitute these and analogous values in (12.22), we get —

yis 5 — — — —j-H

(27r)*aiCT2 . . . CTn-v/w

where

^1 ^2

+ • • •

. . .

This is a form which is very frequently quoted.

( 12 . 26 )

PARTIAL CORRELATION

305

12.29 From these formulae several important results follow immediately.
In the first place, for any fixed values ^2 • • • of ^2 • • • the
exponent (12.25) becomes —

-2r

12 34 . n~

2 ^

ln.2

(n—

-'1 23

O'! 23 rPn,X . . (n-l)

+ constant terms

^12.3

^in 2

23 n

0*2 13

■*

(n-l)^n

(«-.)

+ constant terms.

Hence % is distributed normally about the mean, given by

O’! 23 . . n 0‘2 i3 . . . n i , , (n_l)

(12.27)

Hence every array of every order is normally distributed.

It follows in a similar way that any linear function of the x*s is dis-
tributed normally.

In particular, all deviations of any order and with any number of
suffixes are normally distributed.

12.30 Secondly, as will be seen from (12.27), the regression of on
the other variables is linear. It follows that the regression of any variate
on any or all of the others i^ linear. In (12.27), for instance, the ex-

pressions etc,, are the partial regressions 3 _ etc.

°’2 13 . n

12.31 If, in equation (12.23) any fixed values be assigned to X3 ^2 and
all the following deviations, the correlation between % and Xg, on ex-
panding is, as we have seen, normal correlation. Similarly, if any
fixed values be assigned to to ^^41 23, and all the following deviations, on
reducing fo the second order we shall find that the correlation between
^2 1 and X3 4 is normal correlation, the correlation coefficient being ^23.1* ^ind
so on. That is to say, using k to denote any group of secondary suffixes, (1)
the correlation between any two deviations x^n h ^n,Tc normal correlation ;
(2) the correlation between the said deviation is r^^ whatever the particular
fixed values assigned to the remaining deviations. The latter conclusion, it
will be seen, renders the meaning of partial correlation coefficients much
more definite in the case of normal correlation than in the general case. In
the gjeneral case ^ represents merely the average correlation, so to speak,
between x^^t, and Xn,h * in the normal case r^n.ic is constant for all the sub-
groups corresponding to particular assigned values of the other variables.
Thus in the case of three variables which are normally correlated, if we
assign any given value to Xg, the correlation between the associated values
of % and is r^a.s : in the general case s. if actually worked out for the
various sub-groups corresponding, say, to increasing values of would
probably exhibit some continuous change, increasing or decreasing as the
case might be.

3o6

THEORY OF STATISTICS

12.32 It will be noticed that all the preceding work in this chapter
assumes the correlations to have been determined by the product-sum
formula. The method has also been applied to correlations obtained in
other ways, e,g. from four-fold or contingency tables. In spite of the
favourable results of an experimental test (Newbold, Biometrika, 1925, 17,
251) this procedure remains of doubtful value.

12.33 It has been shown, however, that for the rank correlation coefficient
T a meaning can be assigned to partial coefficients calculated by a formula
analogous to (12.13) for three variables, e.g., for three rankings 1, 2, 3,
we have —

'^12 3 —

^23

(J2.28)

expressing the relationship between rankings 1 and 2 if the influence of
ranking 3 is eliminated. No similar results are known for Spearman's p.

SUMMARY

1. The regression equation of x-^ on x^ , . , x^is written —

% = 612.34 . . n^2 4*6i3 24 . . . n^ 3 "h * • • 23 . . (n-l)^n

The deviation 23 . . . « is defined as

% ^12.34 . . . n ^2 ^ 13.24 . . . • * • ^ln .23 • • • (n~l)^n

and 0*1 23 ... n is the standard deviation of x-^^^ „.

2. The equations giving the regression coefficients are —

^{x^l 23 n)

2 (^ 3 ^ 1. 23 n)=^

23 .n)=0

and similar equations with x^ 13 . «, etc.

3. The product-sum of any two deviations is unaltered by omitting any or
all of the secondary subscripts of the first, if, and only if, all the secondary
subscripts of the first occur among the secondary subscripts of the second;
conversely, the product-sum of any deviation of order p with a deviation
of order ^ the p subscripts being the same in each case, is unaltered by
adding to the secondary subscripts of the former any or all of the q
additional subscripts of the latter.

, — 34 n

12.34 , . 34 . . .

4 .

2 34 ... n

PARTIAL CORRELATION

307

5. Any standard deviation of order p can be expressed in terms of a
standard deviation of order ;5>—l and a correlation of order 1. In fact.

23 . . n— 23 . . . (n-l)(l 23 . , (n-l))

cr| 23

where o) is the determinant

1 ^12 . . . fin

fai 1 fas • ■ ■ ^2n

I »'nl >'n 2 >'«3 • • • 1

and is the minor of the element in the ^th row and the ^th column.

7. Any regression of order p may be expressed in terms of regressions
of order p—\. In fact.

12 34 .

_^12.34

. n

. . (n-l)~~"^ln 34 (n-l)^n2 34

1 34 (n-l)^n2 34. . (n~l)

(n-l)

8. Similarly, for correlations —

_ ^12 34 . (w-l)~~^ln84 n-l)^2n 34 .(n-l)

‘ ‘ ” (1 ■~“^in 34 . . (n-l))Hl ““^2n.34 . . . (n-l))^

9. The coefficient of multiple correlation _ «) is given by

28 . . . n=<^l(l"~«^l(23 . . . n))

;f = l-i^!(23...n)

^31

Also,

1 — ^1(23 . . n)~(l ^12)(^ ^13 2)(^ ^14.23) . . . (^ ^in.23 . . . («-l))

10. i? is necessarily not less than zero. If it is zero, the variate to
which it refers is completely uncorrelated with the other variates. If
jR=l, there is a linear relation between the variates.

11. The multivariate normal surface may be written —

JVl2 . . . n”

-e-H

where

(271)203,0-2 • • •

A ^1/1 J j I Of. ^1^2 j 1 Of.i ^n^n-l

3o8

THEORY OF STATISTICS

EXERCISES

12.1 (Hooker,/. R, Stat, Soc. 1907, 65, 1). The following means, standard
deviations and correlations are found for

Zi=Seed-hay crops in cwts. per acre,

Xg —Spring rainfall in inches,

Xg —Accumulated temperature above 42“^ F. in spring,

in a certain district of England during twenty years.

28*02 <j^= 4.42 4-0*80

^2== 4*91 o-g— 1*10 ri3=-— 0*40

M3=594 0-3=85 ra3=-0*56

Find the partial correlations and the regression equation for hay-crop on
spring rainfall and accumulated temperature.

12.2 In Exercise 12.1, find the multiple correlation coefficient of each
variate on the other two.

12.3 (The following figures must be taken as an illustration only : the
data on which they were based do not refer to uniform times or areas.)

Xi=Deaths of infants under 1 year per 1,000 births in same year (in-
fantile mortality).

Xy —Number per thousand of married w'omen occupied for gain.
X 3 =Death-rate of persons over 5 years of age per 10,000.

X 3 =Number per thousand of population livmg two or more to a room
(overcrowding).

Taking the figures below for thirty urban areas in England and Wales,
find the partial correlations and the regression equation for infantile
mortality on the other factors.

Ml =164 ai= 20*0

M2=158 a2= 74*9

M3=143 0-3= 22*4

M4=205 <74=130*0

12.4 In Exercise 12.3, find the multiple correlation coefficient of on
and Xg ; and of Xi on the other three variates.

12.5 (Data from W. F. Ogburn, '' Factors in the Variation of Crime
among Cities,'* Jour, Amer, Stat, Assoc,, 1935, 30, 12).

For certain large cities in the U.S.A. —

Xi=Crime rate, being the number of known offences per thousand of
population.

Xg— Percentage of male inhabitants.

X 3 = Percentage of total inhabitants who are foreign-born males.

^12 — +0*49 /'ga — +0*15

fi3 = +0*78 f24==-0*37

?'i 4 = +0*20 fg4=+0*23

PARTIAL CORRELATION 309

X 4 =Number of children under 5 years of age per thousand married
women between 15 and 44 years of age.

A's —Church membership, being number of church members 13 years
of age and over per 100 of total population 13 years of age
and over.

M^= 19*9 0 ^= 7-9 ^i 2 = +0*44

Afa- 49*2 02 = 1-3 ri3=-0*34 rgg^-O-SS

10*2 03 == 4-6 y,^==-0*31 r34=+0-44

04=74-4 y,5 = -0*14 r35=+0-33

M5=41‘6 cr5=10-8 r23 = +0 - 25 ^^5=4-0-85

Find the regression equation of on the other four variables. Find also

^1(2346)*

Find, further, 3 , and / 15 . 34 . Discuss the influence of church
membership on crime for these data.

12.6 Show that for n variates there are «C 2 total correlation coefficients,
2) 2 correlation coefficients of order 1 , ^^ 62^0 2 correlation coefficients

of order 2 , and^-^Cg'^Ca of order s. Hence show that there are n{n—l)2*^^
correlation coefficients and n{n'-l)2^-'^ regression coefficients.

12.7 Find the number of multiple correlation coefficients of order s and
the total number of such coefficients for n variables.

12.8 If ail the coirelations of order zero are equal, say=r, what are the
values of the partial correlations of successive orders ?

Under the same conditions, what is the limiting value of r if all the equal
correlations are negative and n variables have been observed ?

12.9 Write dowm from inspection the values of the partial corr<^lations for
the three variables

Zi, Z 2 , and X^=^aX^+bX^

12.10 If the relation

axj.-{~bx2+cx^^0

holds for ail sets of values of x-^, X 2 and x^, what must the partial correlations
be?

CHAPTER THIRTEEN

CORRELATION AND REGRESSION

SOME PRACTICAL PROBLEMS

13.1 The student should be careful to note that the coefficient of correla-
tion, like an average or a measure of dispersion, only exhibits in a summary
form one aspect of the facts on which it is based. Some very real difficulties
arise both m the selection of variables for which the coefficient is to be
computed and in the interpretation of the results when obtained. In
the present chapter we shall consider some of these practical problems
and indicate how they mould from the outset the scope and nature of
an inquiry based on correlations and regressions.

The modifiable unit

13.2 Table 13.1 shows, for each of the 48 agricultural counties of
England in 1936, the yields per acre of wheat and potatoes. The order
of arrangement is the one given in the official Agricultural Statistics.

It is a natural and meaningful question to ask whether there is any
correlation between these 5 delds, so that, for example, we may know
whether an area of high wheat-yield is also one of high potato- 5 deld.

Taking the values of Table ikl as they stand we find a correlation of
+0*2189, a value which the student can verify for himself as an exercise.
But we observe that these yields per acre are given for 48 geographical
areas the boundaries of which are quite arbitrary so far as crop yields
are concerned. WTiat would happen if we took other geographical areas ?
Should we get the same correlation or not ?

We can explore this question to some extent by combining the areas
as given. Suppose we group the counties in pairs and determine for each
of the 24 resulting pairs the simple arithmetic mean yields as exemplified
in the figures following Table 13.1 on the next page.

Since most of the areas are contiguous this is the kind of result
we might get if larger areas 'than counties were recorded. The yields
per acre so calculated are not necessarily those of the grouped pairs
because the total yields may be greater in one member of the pair than in
the other ; but the process will serve for the purposes of illustration.

There are now 24 members and the correlation between the yields will
be found to be +0*2963 against +0*2189 for the original 48. If we
repeat the process and group our 24 pairs (in order as they stand) we find
for the resulting 12 members a correlation of +0*5757. In practice we

3X0

CORRELATION AND REGRESSION

3II

should not compute a correlation for a smaller number of values but if
we pursue the condensing process to the bitter end and group our 12
values into 6, we find a correlation of +0 • 7649 ; and finally, by grouping
the six into three, we have a correlation of +0-9902.

TABLE 13.1. — ^Yields of wheat and potatoes in 48 counties in England in 1936

County

Wtieat
(cwts,
per acre)

Potatoes
(tons
per acre)

County

M^heat
(cwts
per acre)

Potatoes
(tons
per acre)

Bedford

16-0

5-3

Northampton

14-3

4-9

Huntingdon

16-0

6*6

Peterborough

14-4

5*6

Cambridge

16*4

6*1

Buckingham

15*2

6*4

Ely

20*5

5*5

Oxford

14-1

6*9

Suffolk, West

18-2

6*9

Warwick

15*4

5-6

Suffolk, East

16*3

6-1

Shropshire

16-5

6*1

Essex

17*7

6*4

Worcester

14 2

5*7

Hertford

15-3

6*3

Gloucester

13-2

5 0

Middlesex

16-5

7*8

Wiltshire

13*8

6*5

Norfolk

16-9

8*3

Hereford

14 4

6-2

Lincoln (Holland)

21-8

5*7

Somerset

13*4

5-2

,, (Kesteven)

15*5

6 2

Dorset

n-2

6*6

,, (Lindsey)
Yorkshire

15*8

6 0

Devon

14*4

5-8

16*1

6*1

Cornwall

15*4

6-3

(East Riding)
Kent

18-5

6*6

Northumberland

18*5

6*3

Surrey

12*7

4 8

Durham

16-4

5*8

Sussex (East)

15-7

4*9

Yorkshire (N.R }

17*0

5*9

Sussex (West)

14-3

5*1

(W.R)

Cumberland

16*9

6*5

Berkshire

13-8

5-5

17*5

5 8

Hampshire

12-8

6*7

Westmorland

15-8

5*7

Isle of Wight

12-0

6*5

Lancashire

19*2

7*2

Nottingham

15-6

5*2

Cheshire

17-7

6*5

Leicester

15*8

5-2

Derby

15-2

5*4

Rutland

16*6

7*1

Stafford

17-1

6 3

Wheat [cwis,)

Bedfordshire and Huntingdonshire 16*0

Cambridgeshire and Ely 18*45

Suffolk West and Suffolk East ... 17*25

Potatoes {tons)
5*95
5*80
6*5

13.3 We have thus found correlations ranging from 0-2189 to 0*9902.
Nor is this ail. We may well expect that if our 48 counties were divided
into smaller areas the resulting correlation would be smaller than 0-2189.
On the face of it we seem to be able to produce any value of the correlation
from 0 to 1 merely by choosing an appropriate size of the unit of area for

.which we measure the yields. Is there then, any "'real"' correlation
between wheat and potato-yields or are our results illusory ?

13.4 This example serves to bring out an important distinction between
two dMerent types of data to which correlation analysis may be applied.

312

THEORY OF STATISTICS

The difficulty does not arise when we are considering the relationship,
say, between heights of fathers and sons. The ultimate unit in this case
is the individual father or son whose height is a unique non-modifiable
numerical measurement. We cannot divide a single pair of father-and-
son into smaller units ; nor can we amalgamate two pairs to give measure-
ments of the same type as that of the single pair. The same is true of
the data of Table 9.1 (correlation between measurements on shells), of
Table 9.2 (correlation between ages of husband and wife), and of Table
9.4 (correlation between age and weekly milk-yield of cows) — the shell,
the married couple and the cow are non-modifiahle units.

13.5 On the other hand, our geographical areas chosen for the calculation
of crop 3 delds are modifiable units, and necessarily so. Since it is impossible
(or at any rate agriculturally impiacticable) to grow wheat and potatoes
on the same piece of ground simultaneously we must, to give our investiga-
tion any meaning, consider an area containing both wheat and potatoes ;
and this area is modifiable at choice. A similar effect arises whenever
we try to measure concomitant variation extending over continuous
regions of space or time. For example, a regional death-rate must
necessarily relate to a modifiable geographical area ; and rainfall, regional
prices, production of goods or services are quantities of the same type.
In the case where observations are taken over time, examples are imports
and exports, cost of living, and stock-exchange prices. Suppose, for
instance, that we are interested in a possible relationship over time between
the marriage-rate and the wholesale price index, the suggestion being that
in prosperous times, when the price index is relatively high, more people
can afford to marry. Are we to correlate figures compiled on a monthly
basis, a quarterly basis, an annual basis or a triennial basis ? The unit
of time is essentially modifiable.

13.6 From the example we have given as to crop- yields it will be clear
that the magnitude of a correlation will, in general, depend on the unit
chosen if that unit is modifiable. Our correlations will accordingly
measure the relationship between the variates for the specified units chosen
for the work. They have no absolute validity independently of those
units, but are relative to them. They measure, as it were, not only the
variation of the quantities under consideration, but the properties of the
unit-mesh which we have imposed on the system in order to measure it.

13.7 The student should not now go to the other extrerne and claim
that, since a large raiige of values of correlation coefficients may be
obtained according to the choice of a modifiable unit, a particular value
has no signifiance and that any inquiry based on correlations in the
modifiable case is useless. It is of some significance to know that the
correlation between wheat- and potato-yields in the 48 counties of England
in 1936 was 0-2189. A comparison of a series of such values over a
period of years might well throw light on changes in farm practice or

CORRELATION AND REGRESSION

313

soil fertility ; the correlation and the corresponding regression indicates
how far we may expect to predict the potato crop from a knowledge of
the earlier-harvested wheat crop— in tins particular case, not very far.
But we must emphasise the necessity, in this type of work, of not losing
sight of the fact that our results depend on our units. The point assumes
particular importance ''when we are trying to disentangle causal factors.
It is a fact that wheat- and potato-yields in the 48 counties of England
were correlated in 1936 ; but it is a geographical as well as an agricult aral
fact. We cannot infer without additional inquiry that soil which produces
good crops of wheat tends to produce good crops of potatoes.

The attenuation effect

13,8 There is a distinct type of grouping-effect in correlation analysis
which leads to a very similar increase in correlations with increasing
size of geographical area. Suppose we are interested in the relationship
betw^een income and size of family in a certain country. Ignoring minor
difficulties as to what constitutes a family in some cases, we have a non-
modifiable unit. If time, patience and money were available in sufficient
quantity we might be able to ascertain the income and familj^-size for
each unit in the country ; but in practice (unless we performed an ad hoc
sampling inquiry) we should probably have regard to totals and averages
available for regions and districts. We might, for instance, attempt to
estimate the mean number per family for census districts and estimate
the mean income from fiscal or local taxation data. Effectively we should
then be grouping the non-modifiable units into larger units which are
themselves, within limits, modifiable.

13.9 Suppose we have two variables x, y each of which can be regamded
as the sum of a systematic and a random element

x^g+e

y^v+f

(13.1)

We may, for example, imagine that there is some causal factor affecting
^ and 17 simultaneously and hence resulting in a correlation between .v
and y ; but that other components e and / are unrelated to ^ and
and to each other.

Without loss of generality we may suppose that ^ and e are measured
about their means, in v/hich case a; will also be measured about its mean.
We then have

2:(;c2)=S(g2)+2E(g6)+S(^2)

and since ^ and e are uncorrelated we have, on dividing by the number
of the population

var jc==var £-f var e . . . (13,2)

where we write var x fox the variance of r. Equation (13/2) is a particular

314

THEORY OF STATISTICS

case of a theorem which we shall consider in more detail in the next chapter
(14.2).

Similarly we shall have

vary==var ^+var / ... (13.3)

and, writing cov {x, y) for the covariance of x and y

COY [X,y)=cov . . . (13.4)

Let us now denote the correlation between x and y by r and that between
i and by r\ We then have

^ cov(x,y)

{var X var y}^

cov (g, V)

{(var ^+var e) (var ^+var/)P

cov (g, ^)
(var g var 7]Y

(13.5)

Now a variance is essentially non-negative and hence each part of the
denominator on the right hand side of (13.5) is greater than unity. Con-
sequently r is less than r' ; that is to say, a correlation calculated from
the observed values is reduced, or we may say attenuated by the effect
of the factors expressed by e and /.

13.10 Now suppose that we group units, bearing and y values, either
geographically or in time. In virtue of a sampling effect which we shall
study later (Chapter 17) the proportionate variance var ^/var ^ will be
reduced. For the present we assume this ; but the reader will probably
accept it as probable from the consideration that systematic effects
represented by ^ and )} will be cumulative, whereas random effects
represented by e and / tend to cancel out — the larger the number of units
we group, the less, relatively speaking, will their total be affected by
erratic fluctuations.

It foIlow’s that the denominator in (13.5) will also be reduced as we
increase the size of the grouping ; and consequently, if r' is constant r
will continually increase as we group more and more individuals.

13*11 This is the kind of effect we frequently find. It is not necessarily
due to the system which we have just discussed, though that system
provides a possible explanation. There may be other effects such as

patchiness ” in the total area under consideration, which would lead

CORRELATION AND REGRESSION 3 I 5

to Y* itself changing with increased grouping and might either enhance
or counteract the effect of grouping on random components. What
explanation we seek in individual cases depends on the individual cir-
cumstances. We can only leave the reader with the warning to watch
very carefully the possibility of grouping effects, particularly in economic
investigations.

Example 13.1 — (Gehlke and Biehl,/. Am. Stat Ass. Supp, 1934, 29, 169)
A study was made of the relationship between male juvenile delinquency,
expressed as absolute numbers, and the median monthly rental in Cleveland,
Ohio. The 252 census tracts v/ere grouped successively into 200, 175,
150, 125, 100, 50 and 25 areas, consisting so far as possible of the same
size and comprising contiguous territory.

The correlation coefficients, including that for the original 252 tracts, ran
-0-502, -0-569, -0-580, -0*606, -0-662, -0-667, -0-685, -0-763.
The characteristic increase of correlation with size of area is clear. The
corresponding correlations between rates of male juvenile delinquency
and median monthly rentals were —0*516, —0-504, —0*480, -0-475,
—0*563, —0*524, —0*579, —0*621. Here the increase is not uniform
but it begins to appear as the grouping becomes more condensed.

TABLE 13.2. — Numbers of wireless receiving licences issued during the year in the
U.K. and numbers of notified mental defectives in England and Wales
(Date from Statistical Abstract for the United Kingdom. Cmd. 5903, 1939)

Year

Number of wireless
receiving licences
issued (thousands)

Number of notified
mental defectives per
10,000 of estimated
population

1924

1,350

1925

1,960

1926

2,270

1927

2,483

1928

2,730

1929

3,091

1930

3,647

1931

4,620

1932

5,497

1933

6,260

1934

7,012

1935

7,618

1936

8,131

1937

8,593

Note : The year for the purposes of the wireless licence records is
the fiscal year Apnl/March ; for the mental defective records
the census date is January 1st.

Nonsense correlations

13.12 In Table 13,2 we show the number of wireless receiving licences
taken out from 1924 to 1937 in the United Kingdom and the number of

3i6

THEORY OF STATISTICS

notified mental defectives per 10,000 in England and Wales for the same
period. A glance at these figures shows that they are very highly
correlated. The correlation coefficient is, in fact, 0-998.

iNow, facetiousness apart, it cannot be contended that listening to the
radio conduces to notifiable mental defect or vice-versa. The correlation
appears to be nonsensical. Before dismissing it as such, however, we
must concede that the possibility of causal connection cannot be entirely
excluded. For instance, it might be argued that the period m question
was one of great technical progress in many scientific fields ; that one
effect of this movement was the development of broadcasting and the
genoial spread of the practice of listening evinced by the increased number
of licences taken out ; that another effect was the greater interest in
psychological ailments and increased facilities for treatment, resulting
m eithei more discoveries of mental defect or greater readiness to submit
cases to medical notice. Whether this is the right explanation is doubtful,
but it is a possible rational explanation of what at first sight seems absurd.

13.13 The more reasonable explanation is that the strength of the
correlation is an accident ; and our point will have been made if the
reader understands what sort of an accident it is. When we consider
sampling in Chapter 16 et seg. shall discuss the nature of sampling
distributions and shall point out that occasionally, by sheer chance, an
improbable event may arise. In sampling from a bivaiiate normal
population, for instance, as we have pointed out above (9.28) a high
correlation may appear even when the parent is uricorrelated, albeit
rather rarely. This, however, arises in sampling where members are
chosen independently. In the case of our nonsense-correlation we have
taken a sequence of values moving through time, each very dependent on
the one before. Our present effect, accordingly, is not a sampling fluctua-
tion as ordinarily understood.

13.14 It may, none the less, be regarded as accidental. Suppose we
have two series in time, each of which is moving fairly steadily upwards or
downwards (i.e. increasing or decreasing more or less uniformly from
one year to the next). Clearly such series will appear as highly correlated,
positively or negatively, if we happen to chpose for cbnsideration periods
of time in which the movement of each series is in the same direction.
But the reasons for the movements may be quite unrelated or at least
so remote that we cannot claim any ''real'' connection between the two
series. Increased numbers of radio licences are due to the invention of
radio communication and the steady movement towards the saturation of
a latent demand. This is probably quite unrelated to the development
in notifications of mental defectives. It may well be that in a future
period the numbers of licences may decline with a declining population
v/hile the numbers of notified defectives increase.

13.15 It is possible to have nonsense-correlations in space as well as in

CORRELATION AND REGRESSION

327

time, though good examples are hard to find. As we move from north
to south across Europe, for example, the proportion of Roman Catholics
in the population probably increases — there are few in Scotland and a
great many in Sicily. At the same time we should probably find a decrease
in the average height. If, therefore, we were to correlate height and
proportion of Catholics (we have not tried the experiment) we should
probably find quite a substantial negative correlation ; but if so it would
be obvious nonsense in our present usage of the word.

Variate-differences

13.16 Figure 13.1 shows, for the period 1838-1914, the movements of (a)
the infantile mortality (deaths of infants under one year of age per 1,000
births m the same year) and (b) the general mortality (deaths at ail ages
per 1,000 living) in England and Wales. A very cursory inspection of
the diagram shows that the two varied together — when the infantile
mortality rose from one year to the next the general mortality did the
same, with only seven or eight exceptions to the rule during the whole
period under review. The correlation between the annual values of the
two may be expected to be positive, because the infantile death-rate
forms part of the general death-rate ; but it would not be very high
as the general mortality fell more or less steadily from 1875 onwards
w'hereas the infantile mortality rose to a peak in 1898. During a long
period of time the correlation may nearly vanish, for the two mortalities
are affected by largely different causes. In this sense, a high correlation
for a short period might be ‘'nonsense'’ (though this is stretching our
usage rather far) if it was interpreted as implying a strong causal nexus
in the long run.

13.17 To exhibit the closeness of the relation between infantile and

general mortality /or such causes marked changes from one year

to the next it will be best to proceed by correlating the annual changes,
and not the annual values. The work would be arranged in the following
form (only sufficient years being given to exhibit the pnnciple of the
process), and the correlation worked out between the figures of columns
3 and 5 —

Year

Infantile
mortality per
1,000 births

Increase or
decrease from
year before

General
mortality per
1,000 living

Increase or
decrease from
year before

1838

159

22-4

1839

151

-8

21*8

-0-6

1840

154

+3 1

22*9

+ M

1841

145

-9

21*6

-1*3 ‘

1842 i

152

+7 i

21-7

+ 0-1

1843

150

i .-2

21'2

— 0'5

3lS THEORY OF STATISTICS

L Iiifantde mcrtahtfi per 1000 Orths {upper curve)

COHRELATION AND REGRESSION

319

For the period to which the diagram refers, viz. 1838*-1914, the follow-

ing constants were found by this method —

Infantile mortality, mean annual change 0-71
,1 I, , standard deviation 10*76

General mortality, mean annual change — 0*11
„ „ , standard deviation 1*13

Coefficient of correlation -f 0*69

This is a much higher correlation than would arise from the mere fact
that the deaths of infants form part of the general mortality, and con-
sequently there must be a high correlation between the annual changes in
the mortality of those who are over and under 1 year of age, respectively.

13.18 The procedure of the foregoing section has been called the variate-
difference correlation method.*' By taking first differences instead of
the variate values themselves, the slower changes of the two variates
vdth time are to some extent eliminated, and we are able to study the
effect of short-term variations. To eliminate the secular changes more
completely it may be desirable to proceed to second differences, i.e. to work
out the successive differences of the differences in column 3 and column 5
before correlating. It may even be desirable to proceed to third, fourth
or higher differences before correlating. The method should, however, be
used with caution in such cases, particularly with short series. Correlation
coefficients obtained from higher differences are not always reliable, and
their interpretation becomes a matter of considerable difficulty. We
return to the subject later in Chapters 26 and 27 on time-series, where will
also be found a method more adapted to the case of time-series in which
wave-like oscillations appear to be imposed on the general trend.

13.19 When an inquiry involving correlation or regression analysis is
undertaken the variables to be considered are sometimes determined at
the outset by the nature of the questions which are to be answered. If,
for example, we are asked to investigate the relationship between the
annual suicide rate and the annual number of bankruptcies in a particular
country our variables are specified and all that remains is to obtain the
data and to work on them. There may, indeed, be practical difficulties
in obtaining the data for the right years or the right areas but this is not
a matter in which theoretical considerations can help us.

13.20 More usually, the type of inquiry we are asked to undertake is
less definitely specified. We may wish to investigate the relationship
between a number of quantities or factors which are not directly measure-
able, e.g. the relation between weather and the prevalence of epidemic
disease. There is no single measurement corresponding to weather **
and we have to select a number of variables to represent it such as tempera-
ture, rainfall, or cloudiness. Each of these, in general, may be modifiable
or non-modifiable and we have an additional element of choice in the
precise form of the variate which we select.

320

THEORY OF STATISTICS

iS<.21 lii the extreme case we may not even know which factors will
emerge from our analysis as important. Suppose we are interested in the
factors which encourage or prevent tuberculosis and attempt to throw
some light on the subject by considering variations in the incidence of
the disease m different areas. V/hat factors are we to select as in-
” ? It is easy to write down a long list of possible factors —
income, overcrowding, rainfall, sunshine, height above sea-level and so
forth. Assuming for the moment that we can measure all these factors,
how far do we have to take them into account, and can we do so without
: endering the analysis quite unwieldy ?

There is no simple answer to these questions. In the remainder of the
chapter we shall give a short account of some of the resources at the
investigator's disposal in particular cases.

A practical example

13.22 Some of the questions which arise are illustrated in an investigation
by Hooker (/. R. Stat, Soc. 1907, 65, 1) into the relationship between the
yield of certain crops (cereals, roots and hay) and the weather.

The material question here was how far crop-yields tn the same area
vary with the weather. Geographical variation was therefore not in
point, and Hooker considered the series of values over a period of years
for a single area. Climatic, soil, and farm-practice conditions vary so
much over the United Kingdom that any attempt to take geographical
variation into account would have complicated the analysis enormously.
By choosing one area we eliminate some of the variables and can con-
centrate on climatic factors. Our gain in simplicity may, of course, be
offset by loss of generality — we cannot assume that our results will hold
good for other areas where different conditions exist. We must also be
careful to ascertain that, even in the area under consideration, our series
of years is not so long that there are material changes which would obscure
climatic effects, such as exhaustion of soil fertility or a switch from arable
to grass farming,

13.23 There then arises the problem of selecting the appropriate area.
The desiderata are (1) that it should be reasonably homogeneous from the
meteorological standpoint and (2) it should be large enough to present
a representative variety of soil. Hooker chose a group of eastern counties,
consisting of Lincoln, Huntingdon, Cambridge, Norfolk, Suffolk, Essex,
Bedford and Hertford, as fulfilling these conditions. The group included
the county with the largest acreage of each of the ten crops investigated
with the single exception of permanent grass.

13.24 Produce statistics for the more important crops of England and
Wales have been issued by the Ministry of Agriculture since 1885. The
figures are based on estimates of yield furnished by local official estimators
all over the country. Estimates are published for separate counties and

CORRELATION AND REGRESSION

321

for groups of counties (divisions), but not for smaller units of area, though
the crop estimators usually submit returns for parishes.

The data in this case are thus provided by the official publications.
Their nature limits the inquiry in space (since we must choose areas based
on counties) and in time (since figures are not available prior to 1885).
We must also assume that the estimates are reasonably accurate. The
field of choice in most economic inquiries is limited by such factors as
these

13.25 Having decided on our crop*hgures we have to consider the weather
factors. The produce of a crop is dependent on the weather of a long
preceding period, and it is naturally desired to find the influence of the
weather at successive stages during this period, and to determine, for
each crop, which period of the year is of most critical importance as regards
weather. It must be remembered, however, that the times of both sowing
and harvest are themselves very largely dependent on the weather, and
consequently, on an average of many years, the limits of the critical period
will not be very well defined. If, therefore, we correlate the produce of the
crop (X) with the characteristics of the weather (Y) during successive
intervals of the year, it will be as well not to make these intervals too short.
It was accordingly decided to take successive groups of 8 weeks, overlap-
ping each other by 4 weeks, i.e. weeks 1-8, 5-12, etc. Correlation coefficients
were thus obtained at 4-week intervals, but based on 8 weeks' weather.

13.26 Finally, we have to decide what measurable characteristics of the
wreath cr are to be taken into account. Prior knowledge suggests that
the two most important are rainfall and temperature. The two provide
quite enough labour for a first investigation.

(a) The rainfall for a particular county is to some extent a modifiable
unit, for no measurements are taken of the total precipitation on a given
area. Hooker took records of weekly rainfall from eight stations within
the total area under consideration and used the average of these figures
as the first characteristic of the weather.

(b) Temperatures were taken from the records of the same stations.
The average temperatures, however, do not give quite the sort of informa-
tion that is required : at temperatures below a certain limit (about, 42®
Fahr.) there is very little growth, and the grov/th increases in rapidity
as the temperature rises above this point (within limits). It Was therefore
decided to utilise the figures for “ accumulated temperatures above 42®
Fahr.," i.e. the total number of day-degrees above 42° during each of the
8-weekly periods, as the second characteristic of the weather , these
" accumulated temperatures," moreover, show much larger variations than
mean temperatures.

Reference should be made to Hooker's paper for a more detailed account
of the inquiry and its results.

322

THEORY OF STATISTICS

Economy in the number of variables

13.27 In the agricultural case we have just considered there was a
large body of prior knowledge available to assist in determining the field
of inquiry and the variables which were likely to give significant and
meaningful results. This is not always the case. In discussing the
geographical variation of mortality our prior knowledge would suggest
considering as independent variates such factors as age-distribution,
propot cion of males and density of population. We could, however,
without difficulty extend the list of possible factors almost indefinitely,
e.g. by including hours of sunshine, wage levels, adequacy of medical
attention and standards of nutrition. In an investigation into the
variation of crime among American cities Ogburn (/. Am. Stat. Ass. 1935,
30, 12) listed no fewer than 26 factors including birth-rate, proportion of
negroes and proportion of foreign-born immigrants, as well as the more
obvious ones such as efficacy of the police system and proportion of males,

13.28 With adequate data and sufficient patience, of course, we can
work out the regression of our variable on all these others. But the
practical difficulties, including those of computation, are prohibitive ;
and sometimes there are theoretical difficulties into the bargain. The
reader who consults some earlier inquiries in which arithmetical en-
thusiasm was not tempered by common sense will find that there are
more variables than observations and that the resulting high correlations
may mean next to nothing. In any case, ten variables are about as many
as can be conveniently managed, and even that number throws a severe
strain on the computer.

13.29 It is therefore necessary at an early stage to economise in the
number of variables —

(а) As in the agricultural example we may limit the scope of the inquiry.
This is what the physicist does in the laboratory by holding other factors
as constant as experimental conditions will allow. By taking a particular
factor as constant (within reasonable limits) we may ignore its effect on
the regression equation. Subject to practical limitations we exclude in
this way those factors which are expected to have the least effect. We
can always bring them into account later one by one if necessary.

(б) Certain of the variables may be grouped and expressed, at least
approximately, in terms of one of them or of some other summarising
coefficient. In considering the relationship between employment and
retail prices, for instance, we need not bring into account as a separate
variate every retail commodity entering into the household budget. An
index of retail prices would probably be quite sufficient. Again, in a
mortality inquiry we might suppose that ability to pay for medical
attention and standards of nutrition were sufficiently closely linked to
wage-levels to justify us in using wage-levels to represent capacity to
pay the doctor’s biUs and to buy enough food.

CORRELATION AND REGRESSION

3«3

(c) As we have already mentioned, we may proceed by selecting two or
three of the most promising variables to see whether the regression line
containing them satisfactorily accounts for the data (as judged, for
example, by the magnitude of the multiple correlation coefficient.) If
it does not we may add further variates until a good fit is obtained.

13.30 To conclude this chapter we may refer to some approaches to the
problem of statistical relationship which have been developed for particular
purposes but are capable of more general application.

A regression equation expresses the '' best linear relationship between
a dependent variable and a set of given independent variables, best "
in this connection being somewhat arbitrarily defined by minimising a
certain sum of squares. Let us look at this geometrically. Given a set
of points in n dimensions where n is the total number of variables, depen-
dent and independent together, we find as the regression of one on the
others that plane which lies closest to the points ; closest ** being defined
so as to minimise the sum of squares of distances from the points to the
place in the direction parallel to the axis of the dependent variate. The
student can picture this situation easily enough in the two- and three-
dimensional case ; and further dimensions, though impossible to imagine
spatially, add nothing new to the principles.

13.31 Now our cluster of points, though specified by means of n variables
and hence in an n dimensional space, may in fact lie, at least approximately,
in a space of fewer dimensions. For instance the cluster of points of
Figure 12.1 (lying in three dimensions) might perhaps lie on a plane or
even on a line. We may, therefore, be able to find new variables, ex-
pressible as linear functions of the old, which represent the data equally
well but require fewer independent variables.

The approach is one aspect of the subject known as factor analysis. It
seeks to isolate, from a complex of variables, a small number of factors
which will account for most of the variation. We cannot give here any
indication of the various techniques which have been developed, mainly
in psychology, to carry out the analysis, for most of them involve advanced
mathematics as weU as some complicated theoretical problems. The
reader who wishes to pursue the subject may refer to Factor Analysis hv
Hoizinger and Harman or to Kendalls A Course %n MuLUvanate Analysts,
1957.

13.32 A somewhat different line of inquiry known as confluence analysis
has been followed by Scandinavian writers, mainly by Ragnar Frisch.
This involves heavy calculations and in effect, depends on working out
all the possible regressions in order to see how far the appearance of a new
variate disturbes the previous coefficients. For some account of the
method see Frisch's Confluence Analysis, 1934 (Oslo) and Reiersol,
Econometfica, 1941, 2, 1.

324

THEORY OF STATISTICS

SUMMARY

1. Units may be modifiable or non-modifiable. For modifiable units
the values of co: relations depend on the size of the units and must be
interpreted accordingly.

2. When units are grouped and correlations calculated from some
summary features of the group, such as a^-^rages, there may be a tendency
for the correlations to increase with the size of the grouping. Conversely
as the grouping becomes finer the coefficients may be attenuated.

3. Correlations for series which are developing in time may be mis-
leadingly high if the series accidentally happen to move togethei.

4. To elucidate short-term variation in time-senes it may be preferable
to correlate changes from one period to the next rather than the actual
values of the series. This conception is the origin of the variate-difference
method which must, however, be used with great caution.

5. In a general inquiry involving correlation or regression analysis
efforts are necessary to economise in the number of independent variables.

EXERCISES

13.1 Examine how far Tables 9.5 and 9.6 are based on modifiable units.

13.2 The following table shows, for the United Kingdom, the population

and the infantile mortality for certain years-

Year

Population

Deaths of infants per 1,000

(000)

births approx at census date

1871

31,485

144

1881

34,885

134

1891

37,733

141

1901

41,459

140

1911

45,222

108

1921

47,123

1931

47,289

Show that the values are correlated. How far would you regard this as
a nonsense-correlation ?

(Data from the Statistical Abstract for the U.K.Cmd. 5908, 1939. The
figures for 1931 exclude the territory now forming Eire but this may be
ignored for the purpose of the example.)

13.3 The following table shows the number of steam ships registered as

CORRELATION AND REGRESSION

325

belonging to the United Kingdom and the receipts from horse-drawn
vehicle-licenses in Great Britain for certain years —

Year

Number ot steam
vessels

Receipts fiom ;
horse-drawn

1924

10,690

140,719

1925

10,526

118,847

1926

10.262

98,459

1927

10,032

80,302

1928

9,959

64,675

1929

9,855

51,199

1930

9,729

40,878

1931

9,529

32,303

1932

9,248

25,700

1933

8,900

21,288

1934

8,622

17,661

1935

8,306

14,481

1936

8,032

11,579

1937

7,702

9,177

Bearing in mind the development of diesel-propelled ships and of the
motor car, consider how far the correlation between these figures may be
regarded as nonsense.

CHAPTER FOURTEEN

MISCELLANEOUS THEOREMS INVOLVING
THE CORRELATION COEFFICIENT

Algebraical convenience of the correlation coefficient

14.1 It has already been pointed out that a statistical measure, if it
is to be widely useful, should lend itself readily to algebraical treatment.
The arithmetic mean and the standard deviation derive their importance
largely from the fact that they fulfil this'requirement better than any other
averages or measures of dispersion ; and the following illustrations, while
giving a number of results that are of value in one branch or another
of statistical work, suffice to show that the correlation coefficient can be
treated with the same facility. This might indeed be expected, seeing
that the coefficient is derived, like the mean and standard deviation, by a
straightforward process of summation.

The standard deviation of the sum or difference of variables

14.2 Let Z 2 be two variables, and Z stand for their sum or difference.
Let z, Xi, denote deviations of the several variables from their

arithmetic means. Then, if

evidently '

Squaring both sides of the equation and summing,

S (^^) +i:{x,^)±2:l(x,x,)

That is, if r be the correlation between x^ and X 2 , and a, 0*2 the respective
standard deviations,

cr^=o-^^+<y^^±2raj^cr2 .... (14.1)

If % and X 2 are uncorrelated, we have the important special case

cr2=a-i2q_j3.^2 .... (14.2)

The student should notice that in this case the standard deviation of
the sum of corresponding values of the two variables is the same as the

326

MISCELLANEOUS THEOREMS

327

standard deviation of their difference. If we write var X for the variance
of X and cov {X, Y) for the covariance of X and Y we may express (14,1)
as

var (X±y)=var X-{- var cov (Z, Y) (14.3)

and (14.2) as

var (Z± Y)=var X+ var Y . . . (14.4)

The same process will evidently give the standard deviation of a linear
function of any number of variables. For the sum of a series of variables
Zj, Z2, . . . Z;^, we must have —

cr2=ai2+a22+ . . . +a-/+2ri2criCT2 +2^13010-3
+ . . . +2r 23^2^3+ • • •

being the correlation between Zj and Zg/ the correlation between
Zg and Z3, and so on.

Influence of errors of observation on the standard deviation

14.3 The results of 14.2 may be applied to the theory of errors of
observation. Let us suppose that, if any value of X be observed a large
number of times, the arithmetic mean of the observations is approximately
the true value, the arithmetic mean error being zero. Then, the arithmetic
mean error being zero for all values of Z, the error, say, S, is uncorrelated
with X. In this case, if be an observed deviation from the arithmetic
mean, and the true deviation, we have from the preceding —

var %=var ^+var .... (14.5)

The effect of errors of observation is, consequently, to increase the standard
deviation above its true value. The student should notice that the
assumption made does not imply the complete independence of X and d : he
is quite at liberty to suppose that errors fluctuate more, for example, with
large than with small values of Z, as might very probably happen. In
that case the contingency coefficient between Z and 8 would not be zero,
although the correlation coefficient might still vanish as supposed.

14.4 If certain observations be repeated so that we have in every case
two measures and of the same deviation x^ it is possible to obtain
the true standard deviation if the further assumption is legitimate that
the errors and <^2 uncorrelated with each other. On this assumption

var

'Z{XtX^)

" N

and accordingly

. (14.6)

328

THEORY OF STATISTICS

(This formula is part of Spearman’s formula for the correction of the
correlation coefficient , cf. 14.6..)

Isifluence of errors of observation on the correlation coefficient
14.5 Let Vj be the observed deviations from the arithmetic means^
X, y the true deviations, and d, e the errors of observation. Of the four
quantities x, y, d, e we will suppose and v alone lo be correlated. On this
assumption

E(A;ijyi)=E(xy) .... (14.7)

It follows at once that

r.jTiJ'ji GtCJy

and consequently the observed correlation is less than the true correlation.
This difference, it should be noticed, no mere increase in the number of
observations can in any way lessen.

Spearman *s theorems

14.6 If, however, the observations of both x and y be repeated, as
assumed in 14.4, so that we have two measures Xi and x^t yi andy^ of every
value of and y, the true value of the correlation can be obtained by the
use of equations (14.6) and (14.7), on assumptions similar to those made
above. For we have —

^ l!t{XiX^l!t{yiy^ ll{XiX2)y^{yiyz)

(14.8)

Or, if we use all the four possible correlations between observed values of
X and observed values of y,

^4 /14QI

Equation (14.9) is the original form in which Spearman gave his correc-
tion formula. It will be seen to imply the assumption that, of the six
quantities x, y, 8^, only x and y are correlated. The correction

given by the second part of equation (14.8), also suggested by Spearman,
seems, on the whole, to be safer, for it eliminates the assumption that the
errors in x and in y, in the same series of observations, are uncorrelated.
An insufficient though partial test of the correctness of the assumptions
may be made by correlating with yi— y^ : this correlation should

vanish. Evidently, however, it may vanish from symmetry without
thereby implying that all the correlations of the errors are zero.

MISCELLANEOUS THEOREMS

339

Mean and standard deviation of an index

14.7 The means and standard deviations of non-linear functions of
two or more variables can in general only be expressed in terms of the means
and standard deviations of the original variables to a first approximation,
on the assumption that deviations are small compared with the mean values
of the variables. Thus, let it be required to find the mean and standard
deviation of a ratio or index in terms of the constants for X^ and

X^, Let I be the mean of Z, and Mg the means of X^ and Xg. Then,

Expand the second bracket by the binomial theorem, assuming that
X 2 IM 2 is so small that powers higher than the second can be neglected.
Then, to this approximation,

N MaL AfiMj ' ■ ^ ® J

That is, if r be the correlation between x-^ and x^, and if Vg —

I=jf{l-rviv,+v,^) .... ( 14 . 10 )

If s be the standard deviation of Z, we have —

Expanding the second bracket again by the binomial theorem, and neglect-
ing terms of all orders above the second —

1 MA

(

M 2

= ~{i+vi^-‘irvjv^+3vi‘)

or from (14.10) —

=^,(*'1 * - 2n-iVa +^ 48 )

which we may also write as

var X. 2 cov (X,, Xf) , var X;, '
var (Xi/X^) =- - ' MiM., M.^ ,

( 14 . 11 )

( 14 . 12 )

330

THEORY OF STATISTICS

Correlation between indices

14.8 The following problem afords a further illustration of the use of
the same method. Required to find approximately the correlation between
two ratios X^ and X^ being uncorrelated.

Let the means of the two ratios or indices be /j, Jg, and the standard
deviations Si, ; these are given approximately by (14.10) and (14.11) of
the last section. The required correlation p will be given by —

NIJ,

Ms* V

Neglecting terms of higher order than the second as before and re*
membering that all correlations are zero, we have —

pSjSg —

* Af3« "

where, in the last step, a term of the order has again been neglected.
Substituting from (14.11) for and $ 2 » we have finally —

(14.13)

This value of p is obviously positive, being equal to 0-5 if ;

and hence even if X^ and X^ are independent, the indices formed by taking
their ratios to a common denominator X^ will be correlated. The value of
p was termed by Karl Pearson the spurious correlation.'' Thus, if
measurements be taken, say, on three bones of the human skeleton, and the
measurements grouped in threes absolutely at random, there will, neverthe-
less, be a positive correlation, probably approaching 0*5, between the
indices formed by the ratios of two of the measurements to the third. To
give another illustration, if two individuals both observe the same series
of magnitudes quite independently, there may be little, if any, correlation
between their absolute errors. But if the errors be expressed as percent-
ages of the magnitude observed, there may be considerable correlation.
It does not follow of necessity that the correlations between indices or
ratios are misleading. If the indices are uncorrelated, there will be
a similar spurious " correlation between the absolute measurements
ZxX^=Xj^ and Z^X^^X^, and the answer to the question whether the

MISCELLANEOUS THEOREMS

331

correlation between indices or that between absolute measures is xnis«
leading depends on the further question whether the indices or the absolute
measures are the quantities directly determined by the causes under
investigation.

The case considered, where A\, Xg, are uncori elated, is only a
special one ; for the general discussion see K. Pearson, Proc. Roy, Soc,
1897, 60, 489. For an interesting study of actual illustrations see J. W,
Brown and others, J, Roy, Stat, Soc,, 1914, 77, 317.

Correlation due to heterogeneity of material

14.9 The following theorem ofers some analogy with the theorem of
2.26 for attributes : If X and Y ate uncorrelated in each of two records, they
will nevertheless exhibit some correlation when the two records are mingled,
unless the mean value of X in the second record is identical with that in the first
record, or the mean value of Y in the second record is identical with that in the
first record, or both.

This follows almost at once, for if are the mean values of X in

the two records, Xg the mean values of Y, N^, the numbers of
observations, and M, K the means when the two records are mingled, the
product-sum of deviations about M, K is —

Evidently the first term can only be zero if or K—Ki, but

the first condition gives —

that is,

Mi-Af.,

Similarly, the second condition gives Ki —Kg. Both the first and second
terms can, therefore, only vanish if or Kj —Kg. Correlation may

accordingly be created by the mingling of two records in which X and Y
vary round different means.

Reduction of correlation due to mingling of uncorreiated with correlated
pairs

14.10 Suppose that observations of x and y give a correlation
coefficient—

Now, let Mg pairs be added to the material, the means and standard devia-
tions of X and^ being the same as in the first series of observations, but the

332

THEORY OF STATISTICS

correlation zero. The value of 't{xy) will then be unaltered, and we shall
have—

Whence

(ni+n2)cx^(yy

U_ %
ri n^+n^

. (14.14)

Suppose, for example, that a number of bones of the human skeleton have
been disinterred during some excavations, and a correlation is observed
between pairs of bones presumed to come from the same skeleton, this
correlation being rather lower than might have been expected, and subject
to some uncertainty owing to doubts as to the allocation of certain bones.
If Ti is the value that would be expected from other records, the difference
might be accounted for on the hypothesis that, in a proportion jr^

of all the pairs, the bones do not really belong to the same skeleton, and
have been virtually paired at random.

The weighted mean

14.11 The arithmetic mean M of a series of values of a variable X was
defined as the quotient of the sum of those values by their number N, or

M=S(Z) IN

If, on the other hand, we multiply each individual observed value of X
by some numerical coefficient or weight W, the quotient of the sum of such
products by the sum of the weights is defined as a weighted mean of X, and
may be denoted by M' ; so that

jL{W)

The distinction between “ weighted ” and '' unweighted means is,
it should be noted, very often formal rather than essential, for the
“ weights may be regarded as actual, estimated or virtual frequencies.
The weighted mean then becomes simply an arithmetic mean, in which
some new quantity is regarded as the unit. Thus, if we are given the means
Ml, Mg, M 3 . , . , oi r series of observations, but do not know the
number of observations in every series, we may form a general average by
taking the arithmetic mean of all the means, viz. S(M) /r, treating the senes
as the unit. But if we know the numbei of observations in every series it
will be better to form the wetghte^ mean 11(NM) weighting each mean

in proportion to the number of observations in the series on which it is
based. The second form of average would be quite correctly spoken of as
a weighted mean of the means of the several series : at the same time, it
is simply the arithmetic mean of all the series pooled together, i.e. the

MISCELLANEOUS THEOREMS

333

arithmetic mean obtained by treating the observation and not the series
as the unit.

14.12 To give an arithmetical illustration, if a commodity is sold at
different prices in different markets, it will be better to form an average
price, not by taking the arithmetic mean of the several market prices,
treating the market as the unit, but by weighting each price in proportion
to the quantity sold at that price, if known, i.e. treating the unit of quantity
as the unit of frequency. Thus, if wheat has been sold in market A at an
average pnce of 29s. Id. per quarter, in market B at an average price of
27s. 7d. and in market C at an average price of 28s. 4d., we may, if no
statement is made as to the quantities sold at these prices (as very often
happens in the case of statements as to market prices), take the arithmetic
mean (28s. 4d.) as the general average. But if we know that 23,930 qrs.
were sold at A, only 26 qrs. at B and 3,933 qrs at C, it will be better to
take the weighted mean

(29s. Id. X 23,930) + (27s. 7d. X 26) + (28s. 4d. X 3,933)

27,889

=29s.

to the nearest penny. This is appreciably higher than the arithmetic mean
price, which is lowered' by the undue importance attached to the small
markets B and C.

14.13 In the case of index-numbers for exhibiting the changes in average
prices from year to year, it may make a sensible difference whether v/e
take the simple arithmetic mean of the index-numbers for different
commodities in any one year as representing the price-level in that year,
or weight the index-numbers for the several commodities according to
their importance from some point of view. If, for example, our standpoint
be that of some average consumer, we may take as the weight for each
commodity the sum which he spends on that commodity in an average
year, so that the frequency of each commodity is taken as the number of
shillings or pounds spent thereon instead of simply as unity. We revert
to this topic in Chapter 25.

14.14 Rates or ratios like the birth-, death- or marriage-rates of a country
may be regarded as w^eighted means. For, treating the rate for simplicity
as a fraction, and not as a rate per 1,000 of the population,

, Total births

Birth-rate of whole country —

Total population

(Birth-rate in each district X population in that district)

S (Population of each district)

i.e. the rate for the whole country is the mean of the rates in the different
districts, weighting each in proportion to its population. We use the
weighted and unweighted means of such rates as illustrations in 14.16
below.

334

tHKOl^Y OF STATISTICS

14.15 It is evident that any weighted mean will in general differ from
the unweighted mean of the same quantities, and it is required to find an
expression for this difference. If r be the correlation betw^een weights and
variables, cr*c. and cTa, the standard deviations and w the mean weight, we
have at once

i:{WX)=^N{Mw+rGu>^x)

whence

M'=M+rc^^ .... (14.15)

That is to say, if the weights and variables are positively correlated, the
weighted mean is the greater ; if negatively, the less. In some cases r is
very small, and then weighting makes little difference, but in others the
difference is large and important, r having a sensible value and jw a
large value.

14d6 The difference between weighted and unweighted means of death-
rates, birth-rates or other rates on the population in different districts
is, for instance, nearly always of importance. For instance, in 1941, the
birth-rates per 1,000 civilian population in Lancashire weie —

County Boroughs ....

le^i

Urban Districts

14-7

Rural Districts

14-4

The mean value of these three is 15*07 w’^hereas the birthrate for Lanca-
shire as a w^hole was 15*5, a reflection of the well-known fact that the
more populous areas have the higher birth-rate. The death-rates, ex-

eluding civilian war-deaths, were —

County Boroughs ...

15*6

Urban Distnets

13*2

Rural Districts

11*0

with a mean of 13-27, against a (weighted) mean for the whole county
of 14-5. There appears to be a positive correlation between death-rate
and size of population as well as between birth rate and population,
though no doubt for different reasons. Urban aggregations have a larger
proportion of the young than^ rural areas, and hence a higher birth-rate,
but on the other hand living conditions are more unfavourable to life
and this factor outbalances the effect of the more favourable age-com-
position on the death-rate.

Age-composition may exert a similar effect on marriage rates. For

MISCELLANEOUS THEOREMS 335

instance, persons married per 1,000 in the regions of England and Wales
in 1941 were as follows —

South East ...

. 21*6

North I

... 19*5

North II

19*0

North III

.... 19*9

North IV .

19*9

Midland I

20*0

Midland II

. . 19*2

East

. . 19*0

South-west

17*2

Wales I

. . 20*1

Wales II

. 16*3

The mean of these figures is 19*25 whereas the marriage rate for the
whole country was 20*1. The explanation is that the more populous
areas contain a greater proportion of younger people and hence have a
higher marnage-rate.

14.17 The principle of weighting finds one very important application
in the treatment of such rates as death-rates, which are largely affected
by the age and sex composition of the population. Neglecting, for
simplicity, the question of sex, suppose the numbers of deaths are noted
in a certain district for, say, the age-groups 0—, 10— , 20— , etc., in which
the fractions of the whole population are ^2, etc., where
Let the death-rates for the corresponding age-groups be etc. Then

the ordinary or crude death-rate for the district is

(14.16)

For some other district taken as a basis of comparison, perhaps the
country as a whole, the death-rates and fractions of the population in the
several age-groups may be 8 2 , ^3, . . ^1, n-g, 773, . . and the crude
death-rate

A=S(^7r) .... (14.17)

Now, D and A differ either because the d's and difer or because
the p's and n's differ, or both. It may happen that really both districts
are about equally healthy, and the death-rates approximately the same
for all age-classes, but, owing to a difference of weighting, the first average
may be markedly higher than the second, or vice versa. If the first
district be a rural district and the second urban, for instance, there will be
a larger proportion of the old in the former, and it may possibly have a
higher crude death-rate than the second, in spite of lower death-rates in
every class. The comparison of crude death-rates is therefore liable to
lead to erroneous conclusions. The difficulty may be got over by averaging
the age-class death-rates in the district not with the weights pi, pp ^>5, . . ,

336

THEORY OF STATISTICS

given by its own population, but with the weights . . . given

by the population of the standard district. The standardised death-rate
for the district will then be

D'=:S(i7r) .... (14.18)

and D' and A will be comparable as regards age-distribution. There is
obviously no difficulty in taking sex into account as well as age if necessary.
The death-rates must be noted for each sex separately in every age-class
and averaged with a system of weights based on the standard population.
The method is also of importance for comparing death-rates in different
classes of the population, e.g. those engaged in given occupations, as
well as in different districts, and is used for both these purposes m the
publications of the Registrar-General for England and Wales.

14.18 Difficulty may arise m practical cases from the fact that the

death-rates d^y d^, d^, . . . are not known for the districts or classes which
it is desired to compare with the standard population, but only the crude
rates D and the fractional populations of the age-classes Pz> * * •

The difficulty may be partially obviated (cf. 2.30 and Example 2.10,
pp. 38-40) by forming what is termed an index death-rate A' for the class
or district, A' being given by

A'=£((Jji>) .... (14.19)

i.e. the rates of the standard population averaged with the weights of
the district population. It is the crude death-rate that there would be in
the district if the rate in every age-class were the same as m the standard
population. An approximate standardised death-rate for the district or
class is then given by

D"=Dx~ .... (14.20)

D" is not necessarily, nor generally, the same as D\ It can only be the
same if

HjdTr) JLjdTT)

t{dp)~i.{dp)

This will hold good if, e.g., the death-rates in the standard population
and the district stand to one another in the same ratio in all age-classes,
i.e. d^jdj^^^d^ld^^d^ldQ^eic. This method of standardisation was used
in the Annual Summaries of the Registrar-General for England and Wales.

14.19 Both methods of standardisation — that of 14.17 and that of
14.18 — are of great importance. They are obviously applicable to other
rates besides death-rates, e.gi birth-rates. Further, they may readily be
extended into quite different fields. Thus it has been suggested that
standardised average heights or standardised average weights of the children

MISCELLANEOUS THEOREMS

337

in different schools might be obtained on the basis of a standard school
population of given age and sex composition, or indeed of given composi-
tion as regard hair- and eye-colour as well.

14.20 In 14.11-14.16 we have dealt only with the theory of the weighted
arithmetic mean, but it should be noted that any form of average can be
weighted. Thus a weighted median can be formed by findmg the value
of the variable such that the sum of the weights of lesser values is equal
to the sum of the weights of greater values. A weighted mode could
be formed by finding the value of the variable for which the sum of the
weights was greatest, allowing for the smoothing of casual fluctuations.
Similarly, a weighted geometric mean could be calculated by weighting
the logarithms of every value of the variable before taking the arithmetic
mean, i.e.

log Gw

I,(W log X)
= ^W)

SUMMARY

1. The standard deviation of the sum of variables Xg, . . .
is given by

. . . +(yN^+2r^^<Ji(T^+2ri^(j^(Js+ • • . . . .

which may also be written

var {S(X)}=S(var X)+S{cov(Xi, X^)},

2. In particular, the variance of the sum of N uncorrelated variates is
the sum of their variances.

X X

3. If X^, Xg and X3 are uncorrelated, the indices •— will neverthe-
less be correlated in general.

4. If X and Y are uncorrelated in each of two separate records, they
will be correlated in the sum of the two records, unless either the means
of X or the means of Y, or both, are the same in the two records.

5. If correlated and uncorrelated material is mingled, the correlation
in the total is lower than that in the correlated portion.

6. An arithmetic mean is weighted when, in the calculation of ^(X),
each value of the variate is multiplied by a weight W.

7. The weighted arithmetic mean is greater or less than the unweighted

according as the weights and variables are positively or negatively
correlated.

338

THEORY OF STATISTICS

EXERCISES

14,1 (Data from the Decennial Supplements to the Annual Reports of the
Registrar-General for England and Wales.) The following particulars
are found for 36 small registration districts in which the number of births
in a decade ranged between 1,500 and 2,500 —

Decade

Proportion of male births
per 1,000 of all births

Mean

Standard

deviation

1881-1890 .

508*1

12-80

1891-1900 .

508-4

10-37

Both decades

508 25

11-65

It is believed, however that a great part of the observed standard
deviation is due to mere “ fluctuations of sampling of no real significance.

Given that the correlation between the proportions of male births in a
district in the two decades is -t-0'36, estimate (1) the true standard devia-
tion freed from such fluctuations of sampling ; (2) the standard deviatioir
of fluctuations of sampling, i.e. of the errors produced by such fluctuations
in the observed proportions of male births*

14.2 The coefficients of variation for breadth, height and length of
certain skulls are 3*89, 3-50 and 3*24 per cent respectively. Find the
"spurious correlation" between the breadth /length and height /length
indices, absolute measures being combined at random so that they are
uncorrelated.

14.3 (Data from Boas, communicated to Pearson ; cf. Fawcett and
Pearson, Proc. Roy. 5oc., 62, p. 413) From short series of measurements
on American Indians, the mean coefficient of correlation found between
father and son, and father and daughter, for cephalic index, is 0*14;
between mother and son, and mother and daughter, 0*33. Assuming
these coefficients should be the same if it were not for the looseness of
family relations, find the proportion of children not due to the reputed
father.

14.4 Find the correlation between X^+X^ and *^3

being uncorrelated.

14.5 Find the correlation between and aX-^-\-hX^, X^ and X^ being
uncorrelated.

14.6 (Referring to 13.17.) Use tlie answer to Exercise 14.5 to estimate,
very roughly, the correlation that would be found between annual

MISCELLANEOUS THEOREMS

339

movements in infantile and general mortality if the mortality of those
under and over 1 year of age were uncorreiated. Note that —

General mortality per
1,000 of population

=Infantile mortality per 1,000 births X

Births

Population

■f Deaths over one year per 1,000 of population

and treat the ratio of births to population as if it were cons tan I at a rough
average value, say 0*032. The standard deviation of annual movements
in infantile mortality is {loc. at.) 10*76, and that of annual mov^ements in
mortality other than infantile may be taken as sensibly the same as that
of general mortality, or, say, 1*13 units.

14.7 If the relation

holds for all values of x^ and (which are, in our usual notation,
deviations from the respective arithmetic means), find the correlations
between x^ and x^ in terms of their standard deviations and the values
of a, h and c.

14.8 What is the effect on a weighted mean of errors in the weights of the
quantities weighted, such errors being uncorrelated with one another, with
the weights or vdth the variables : (1) if the arithmetic mean values of
the errors are zero, (2) if the arithmetic mean values of the errors are not
zero ?

14.9 The following are the variances of the rainfall (1) for January to
March, (2) for April to December, (3) for the whole year, at Greenwich in
the eighty years 1841-1920, the unit being a millimetre —

January-March . . . 0*3^^= 1,521

April-December . . . , a 2 ^= 8,968

Vidiole year .... a^==10,754

Find the correlation between the rainfall in January-March and April-
December.

14.10 If of three variables A, B, C, the variance of the sum of A and B

is the sum of the variances of A and B and the variance of the sum of
B and C is the sum of the variances of B and C ; show that the variance
of the sum of A and C is not necessarily the sum of the variances of A
and C. What must be the correlation between and B+C for

this to be true ?

CHAPTER FIFTEEN

SIMPLE CURVE FITTING

The problem

15,1 In this chapter we turn aside somewhat from the line of development
of previous chapters in order to study a subject of considerable theoretical
and practical importance — the representation of relationship between
two variables by simple algebraic expressions. Our work on correlation
has already led us to fit regression lines and planes to the means of arrays.
We now attack a rather more general problem. An illustration will make
clear the type of inquiry involved.

TABLE 15.1, — Estimated distance and velocities of recession of 10 extra-galactic

nebidae

(Edwm Hubble and Milton L. Humason, “The Veloaty-distance Relation among Extra-galactic Nebulae,”
ContnMfons from Mount Wtlson Observatory, Carnegie Institute of Washington, No. 427 , Astrophystcal Journal,

1931, 74 , 43)

Constellation in
which the nebula
is situated

Mean velocity
(kilometres per
second)

Distance
(millions of
parsecs)

Isolated Nebula II .

630

1-20

Virgo

890

1-82

Isolated Nebula I .

2,350

3*31

Pegasus

3,810

7*24

Pisces

4,630

6*92

Cancer

4,820

9*12

Perseus

5,230

10*97

Coma

7,500

14*45

Ursa Major

11,800

22*91

Leo .

19,600

36*31

Table 15.1 shows the estimated distance and velocities of recession of
certain nebulae in the outlying parts of the visible universe.

A little inspection of the table will show that there appears to be some
relation between distance and velocity — the greater the one, the greater
the other, wdth only one exception. A diagram makes the relation clearer
still. In fig. 15.1 we have taken the two variables velocity and distance
as rectangular co-ordinates y and x, and have marked for each nebula
a point whose co-ordinates are the distance and velocity of that nebula.

' The ten points so obtained evidently lie very approximately on a straight

340

SIMPLE CURVE FITTING 34I

line or, to express the same fact algebraically, the ten values of the variables
are closely represented by an equation of the form

y=ao+a^x .... ( 15 . 1 )

where we use small letters to denote current co-ordinates.

15.2 No straight line, however, passes exactly through all the points,
although a great many lines may be drawn which nearly do so. The
question then arises, is there a straight line which fits the points better
than all others, and if so, which is it ^ Or, in other language, what values
of ao and in equation (15.1) must we take to get the best representation
of the linear relationship betw^een the two vanables ? And, as a further
question, can we devise a measure of the closeness of the fit of the various
lines which can be drawn ?

Distaiv^e (millions of parsecs)

Fig. 15.1. — Relationship between distance and velocity of recession in certain extra-

galactic nebulae (Table 15.1)

15.3 In the foregoing illustration it is clear from the data or from the
diagram that a linear relationship between the variables gives a very
close picture of the truth. In other cases the points of the diagram will
lie more or less on a curve, and no straight line will give a satisfactory
representation. We should then wish to investigate whether the depend-
ence of y on jjt; may be suitably represented by the more general equation

. . . -{-apxP . . . (15.2)

which, in the diagram, corresponds to a curve of the type known as
parabolic. The number p indicates the degree of the parabola, and we
speak of quadratic, cubic, quartic parabolas, meaning curves of type
(15.2) with ^=2, 3, 4, respectively.

342

THEORY OF STATISTICS

15.4 Our general problem may, then be stated as follows : Given n

pairs of values of two variables, to express the

values of one of them as nearly as may be in terms of the other by an
equation oi the form (15.2) ; and to measure the closeness of the approxi-
mation of the values of y given by the equation to the actual values. In
geometrical language, given n points in a plane, to fit to them a curve oi
the parabolic type (15.2) and to measure the closeness of lit.

15.5 The representation of data in this way may serve several purposes.
In the first place, it may present the relationship between the two variables
in a useful summary form. Secondly, it ma}^ be used to interpolate, i.e.
to estimate the values of one variable which would correspond to specified
values of the other. In fig. 15.1, for example, the straight line which
has been drawn in, and whose equation is obtained below, tells us what
we might expect to be the velocity of a nebula whose distance is, say,
20 million parsecs, on the assumption that the linear relation holds good
for nebulae in general.

15.6 Again, the representation may also be very suggestive to the
theorist. The linear form of the relationship between the variables of
Table 15.1 involves more than a convenient summary of the facts, and has
inspired a great deal of research into the nature of the physical universe.
In such cases, the derived equation is regarded as the expression of a law
of nature, and the deviations of the observed values from those given
by it are interpreted as fluctuations arising from experimental error or
secondary perturbations. This standpoint is common in physics, in which
data often lie very closely about a smooth curve.

The method of least squares

15.7 Let us suppose that we have n pairs of values

and that we wish to represent them by an equation of the type (15.2).
Our problem is, having fixed the value of p, to determine the constants
Uq, ^ 3 ^, . , . in terms of the observed values X, Y, so as to get the best
possible fit.

The expression best possible fit may be defined in more than one
way, and consequently there is no unique method of determining the
constants. Several methods have been proposed, and our choice between
them is determined mainly by convenience. One way, which is suggested
by the geometrical representation, is to choose the curve of equation
(15.2) so that the sum of the distances (taken as positive) of the points
from it is a minimum, the sum of the distances being regarded as a measure
of goodness of fit, and the best fit being given by the curve of specified
degree for which that sum is least. But this method, whatever its
theoretical attractions, suffers from the disadvantage that it is difficult
to apply in practice except for the straight line.

An alternative method, which is in almost universal use at the present
time, is that known as the Method of Least Squares, and we proceed to

SIMPLE CURVE FITTING

343

discuss it at length. We have already used it to find regression lines
(9.20 and 12.4).

15,8 If we substitute for the value in equation (15.2) we get a quantity
given by

• (15.3)

This is not in general the same as and we therefore define the residual
gr as

^j. = Yj. . (l0.4)

There will be n residuals, one for each pair X, Y, and they are all zero
if, and only if, the curve is a perfect fit. We then take the sum of the
squares of residuals —

. . (15.5)

If U is zero, each residual must be zero, and the data are represented
perfectly by the equation. Except in this case, V is positive. The
further the points lie from the curve of equation (15.2), the greater V
will be. TJ therefore provides one measure of the closeness of fit. From
this standpoint, the best fit will be that for which X] is least.

The Method of Least Squares adopts this criterion, and states that
the constants a shall he determined so that U is a minimum,

15.9 The reason for taking the sum of squares of residuals, rather than
the, sum of residuals simply, is akin to that which led us to prefer the
standard deviation to the mean deviation as a measure of dispersion
(Chap. 6), namely, that the former is more convenient in theory and leads
to equations which are easier to handle in practice.

15.10 It was formerly the custom, and is so still in works on the theory
of observations, to derive the method of least squares from certain
theoretical considerations, the assumed normality of the distribution of
errors of observations being one such. It is, however, more than doubtful
whether the conditions for the theoretical validity of the method are
realised in statistical practice, and the student would do well to regard
the method as recommended chieti}' by its comparative simplicity and by
the fact that it has stood the test of experience,

15.11 Consider now the quantity XJ, given by equation (15.5),

(3Ei, . . . a^ are to be chosen so that this is a minimum, say Let us
imagine this done.

344

THEORY OK STATISTICS

If, now, we substitute in equation (15.5) <Jo+«o for a^, for Uj.

flj+Cj for a^, and so on, we shall get a quantity given by

l/i=S{Y-(ao+eo)-(«i+ei)^- • • .

and Ui is greater than Uq for all values of ep, e^, . . . e,.

Now,

l\=={i:{Y~ao-aiX- . . . -a,Z^)-{eo+eiX+ . . . +e^P)Y

=S(y-ao-«i^- • • •

-2E(F-ao-«i^- • • • -a,X#)(co+eiit+ . . . +6^.^^)

-!-S(eo+eiX+ . . . -\-ejiXP)^

The first of these terms is equal to U^. Hence, if U-y'^U^, we must have
— 2S(y — a® — ciyX — . . . — aj,X^)(eo+eiX+ . . . +ej>X^)

+2(eo+eiZ+ . . . +e„XP)^^0 (15.6)

This is to be true for all values of . . . e^. Let us then take these
quantities to be very small. The second term in equation (15.6), depend-
ing as it does on the squares of the e's, will be small com.pared with the first,
and may be neglected. (15.6) will then be true only if the first term
vanishes, for otherwise the e’s could be so chosen in sign as to make the
first term negative.

Hence,

S(y-ao-ai-S:- . . • -a^XP)(e^+eyX+ . . . +e„XP)^0 . (15.7)

This is true for all small values of the e’s. Hence the coefficients of
eo, €i, . . . e^, all vanish, i.e, we have —

S(y) -ao« ~ayL{X) - . . . -a^(X<') =0 )

2(yZ) -floS(X) -ayLiX^) - . . . -a,S(X^+i) =0

S(yX*)-aoS(i^'') -aiS(X*«) - . . . -a^{XP+^) =0 , (15.8)

Y{YXP)-a,;S,{XP)-ayi:{XP+^)- . . . -apS(X*#) =0 )

The equations (15.8) give us ^-4-1 equations in the (^.4-1) unknowns
ao . . . a». Hence they may be solved so as to give the a’s in terms of-
the calculable quantities S(A'), S(Z®), . . . S(X®^), S(y), 2(yA'), . . .
^YXP).

15.12 It will be seen that the solution of these equations depends on
the evaluation of the various summed quantities. A first step is therefore
to calculate these sums, and this is done by a process very similar to that
used in finding the moments of a distribution.

SIMPLE CURVE FITTING

345

We can, in fact, express the equations in terms of moments. Dividing
each equation by «, and remembering that we have —

— . . . =0 \

(15.9)

-S ( y p ' — 1 —a 2 /i '3, + a -

Equations for fitting a straight line

15.13 In the simplest case, that of a straight line, we have ^ = 1, and
the equations (15.9) become —

-E(y)

rt>

h{YX)=a,h'+a,,i,'

(15.10)

In particular, if X and Y are measured about their means and hence
are denoted by x, y, we have — ■

/«i=0

S(y)=o

and hence, from (15.10),

ao=0

so that the fitted line is

«i= — 2(3'*)

«/ig

3'=*— 2(y*)
n/i^

(15.111

i.e. passes through the mean of X and y . This is, in fact, the first regression
equation of (9.6) (p. 216) in another form.

15.14 In equation (15.2) it is customary to call x the " independent
variable and y the '' dependent variable. In any given case it is, as a
rule, possible to regard either of the variables under consideration as the
independent variable, and the other as the dependent variable. We shall
then get two expressions, one giving variable A in terms of variable B, the
other giving B in terms of A ; and there will be two curves of closest fit,
just as there are two regression lines in the theory of correlation.

346

THEORY OF STATISTICS

These two curves are not, in general, the same, and the result sounds a
little paradoxical until we examine how the two curves are derived. We
have, in fact, two definitions of closest fit, one minimising residuals of the
type the other minimising residuals of the type

On a grounds there is nothing to choose

between the two,

15.15 Which of the two forms we choose will depend in practice on
a variety of circumstances. Sometimes one variable is clearly marked out
as the independent variable. For example, in considering the way in
which a population varies with time, it is almost inevitable to regard the
former as dependent on the latter, and not vice versa. In other cases the
choice is dictated by the purpose in view. For instance, in expressing the
relationship between current and resistance in an electric circuit, an in-
vestigator would probably take as the independent variable that factor
over which he had direct control. Frequently, however, there is no guide
of this kind, and it may be necessary to ascertain both curves. See 15,27
below.

Calculation

15.16 The calculations necessary to fit a curve by the method of least

squares fall into two stages. First of ail, the sums of squares which
appear in equation (15,8) must be found, or, what amounts to the same
thing, the moments. To fit a curve of degree p it is necessary to find 2p
sums of the type 2(Z*) and ^-{-1 sums of the type E( (including S( Y)).

The work is best carried out systematically after the manner of Chapter 7,
and several devices considerably shorten the arithmetical labour.

{a) ,By a suitable choice of origin and unit we can often reduce the
given values of X and Y to smaller numbers — a great help in calculating
the higher powers and sums. For instance, if the values of Y were 625,
650, 675, 700, w^e could take an origin at y=625, and a scale of one unit
=25, and our new values would then be 0, 1, 2, 3.

(b) If the values of the independent variable proceed by equal steps,
and particularly if there is an odd number of them, the labour of calcula-
tion is enormously reduced. We shall consider this important case in
some detail below (15.22).

When the various sums have been ascertained, the second stage, that
of the solution of the equations (15.8), may be carried through. For a
curve of degree p there are of these equations. They are linear in
the unknowns a, and their solution offers only arithmetical difficulty.

15.17 Before proceeding to consider some examples, we may remark
on one point of theoretical interest. It is always possible to fit a curve
of degree p exactly to points ; for instance, a straight line can be
drawn to pass exactly through two points, a cubic parabola through four
points, and so on. Thus, if we have n points we can always find a curve

SIMPLE CURVE FITTING

347

of degree n — I which is an exact fit. But in practice n is rarely less than
ten, and a fitted curve of degree as high as this would have no practical
value and very little theoretical interest. It is only exceptionally that use
IS found for fitted curves of degree higher than the fourth.

We will nov/ consider some examples.

Example 15.1. — Let us fit a straight line to the data of Table 15.1. To
illustrate the method we will deal with both cases, taking first distance and
then velocity as the independent variable.

Denoting, then, distance by .r and velocity by y, we wish to fit a curve
of the form

For this we require S(W), S(A"“), E(Y) and II(YA). For the alternative
case we shall also require S(y2).

The arithmetic is shown in Table 15.2. In successive columns we write,
for each nebula, Y, A, YA and Y^. Totals are shown at the foot of
the columns.

TABLE 15.2. — ^Practical work for fitting a straight line to the data of Table 15.1

Constellation

Mean velocity
(000 km per
second)

Distance
(millions of
parsecs)

A'2

Isolated Nebula II

0 63

1-20

1-4400

0 7560

0 3969

Virgo

0 89

1-82

3*3124

1*6198

0*7921

Isolated Nebula I

2-35

3-31

10*9561

7*7785

5*5225

Pegasus.

3-Sl

7-24

52*4176

27*5844

14*5161

Pisces .

4*63

6-92

47*8864

32 0396

21*4369

Cancer

i 4-82

9-12

83*1744

43*9584

23*2324

Perseus .

5 23

10*97

120*3409

57*3731

27*3529

Coma

7 50

! 14 45

208*8025

108*3750

56*2500

Ursa Major

11 80

1 22 91

524 8681

1 270*3380

139*2400

Leo

19-60

36*31

1318*4161

711 6760

384*1600

Total

61 26

: 114*25

2371*6145

4261*4988

672*8998

Equations (15.8) then become

E(Y)-^io>^--aiS(A)=0

S( YA) -a^[X) -^iSiA^) =0
or

61 =0
1261 •4988-114*25^0-2371 *6145^^^ =0

iSlultiplying the first of these by 114*25 and the second by 10, and sub-
tracting, we get

5616 • 033-10,663 • 0825^^ =0
45^=0*527 (more accurately, 0*526,680,066)

348

THEORY OF STATISTICS

and hence.

^0=0- 109 (more accurately, 0- 108,680,240)

So that

y=0- 109 +0-527^

{a)

This hne is

shown in fig. 15.1.

If we wish to express distance in terms of velocity, we have,
changing X and Y in equations (15.8) —

inter-

x==a^’+a^'y

S(X)-«o'«-«i'S(F)==0

'Z{XY) -ao'S(y) -ai'S(y2) =0

114-25-10ao'-61-26ai'=0

1261 -4988-61 •26ao'-672-8998ai'=0

whence

ao' = — 0-135
ai'= 1-89

and

x==-0- 135+1 -89^

ih)

Equations {a) and (5) are nearly identical, for dividing (a) by 0*527
and rearranging, we have —

= — 0*207+1 -OOy

This is exceptional, and results from the closeness with which the points
lie to a straight line. The correlation between X and Y is, in fact, 0*997.

Reduction of data to linear form

15.18 Example 15,2. — It sometimes happens that we may reduce data
to a linear form by some simple transformation. Table 15.3, for
example, shows the number of fronds of a duckweed plant on fourteen
successive days. The number of fronds {N) clearly does not increase
uniformly with time {x), and the curve of growth is not linear, as may be
seen by graphing N against x. There are theoretical reasons for inquiring
whether the law of growth may be represented by an equation of the form

A population which conformed to this equation would have the property
that its rate of increase at any moment was proportional to the size of
the population at that moment — its birth-rate,’' so to speak, would be a
constant.

Taking logarithms, we have —

log, iV=log, a+hx

SIMPLE CURVE FITTING

349

and if we now write y N, we have —

jy=log, a^hx

which is linear in x and y.

We should, of course, have a relation of the same form, with different
values of the constants a and 6, if we took logarithms to base 10, which
IS usually the more convenient procedure.

We therefore try the effect of fitting a straight line to a; (the time) and
logio ^ number of fronds). From fig. 15.2 it will be seen that the
fit is a close one.

Fig. 15.2. — Straight line fitted to data of Table 15.3. (Growth of duckweed)

The preliminary work is shown in Table 15.3. We find first F, corre-
sponding to logio iV, then S(X), 2(7), S(X2), 2(yX). For this particular
example we do not require 2(7^). In view of the simple character of
the values of X there is little saving in taking other origins or units for
X and 7, although, if we were fitting a curve of higher order, it might
be an advantage to take a different origin for Z.

350

THEORY OF STATISTICS

TABLE 15.3. — Growth of duckweed

(V H Blackman, Nature, 6th June, 1936, quoting data cf Asnhy and Osley.)

Number of fronds

logic IV

Days

100

2-0000000

127

2*1038037

4*2078074

171

2*2329961

6 6989883

233

2*3673559

9-4694236

323

2*5092025

12*5460125

452

2*6551384

15*9308304

654

2*8155777

49 1

19*7090439

918

2*9628427

23 7027416

1,406

3 1479853 !

1 28*3318677

2.150

3*3324385

10 !

100

33 3243S50

2,800

3*4471580

121

! 37 9187380

4,140

3*6170003

144

43 4040036

5,760

3*7604225

169

1 48*8854925

8,250

3*9164539

196

54 8303546

Total

40-8683755

105

1015

340*9594891

Equations (15.8) then become —

Z{Y)--na^^a^{X)^0
S(yZ) ~-ao2(Z) ^a;L[X^) =0
or

40-8683755-14ao- 105^3=0
340 • 9594891 105^0 ~ 1 01 Sa^ =0

whence

^0=1 -785
ai=:0-1514

and

3/=! .785+0* 1514;^; ... (a)

Raising this to power 10, and remembering that we have —

Ar=:ioi*786x ... (6)

which we may also write, expressing the powers of 10 as actual numbers —

iV=60*95x (1*417)'®

15.19 Example 15.3. — The process of taking logarithms may be applied
to both variables. In Table 15.4 are given the costs per unit of electricity
sold [rj) and the number of units sold per head of the population served
by the undertaking (^) for 27 electricity undertakings. The data were
taken from the Returns of the Electricity Commission for 1933-34, which
cover about six hundred undertakings, by selecting every twenty-fifth.
They are, therefore, only a comparatively small sample, but they reflect
fairly accurately the general relationship between ^ and tj for the whole
number of undertakings.

SIMPLE CURVE FITTING

35 ^

This relationship is illustrated by fig. 15.3, on which ^ is graphed against
Tj, It will be seen that, broadly, the larger the number of units sold per
head, the lower the cost per unit.

The points of fig. 15.3 lie, in fact, about a curve which suggests a
relation of the form —

As ^ becomes larger, 7} becomes smaller, and as ^ tends to zero, 7 tends to
infinity. Let us try to fit a curve of this kind to the data.

We have —

and, putting

log 7j==\og a— blog ^
y=log77, A;=logg

y=log a—bx

which is linear. We therefore proceed to fit a straight line to log r/ and log J.

Fig. 15.3. — Curve fitted to data of Table 15.4

352

THEORY OF STATISTICS

The preliminary work is shoTO in Table 15,4. Equations (15.8) become,

in the usual way,

whence

5-2493 -27flo-50-1311ai=0

7 • 3008-50 -1311 ao-97 • HSOa^ =0

and

ao=l-31 ai=-0'601

From which

:y=f- 31-0- 601:*;

(a)

(6)

17=20- 42^-0

Fig. 15.4 shows the values oiy plotted against those of x. The straight
line we have found cannot be described as a good fit, but so far as the eye
can judge it is as good as any simple curve is likely to be. It expresses
the general relation between x and y ; but, naturally, local circumstances
cause individual values to deviate appreciably from this relation. Statis-
tical data which are not produced under laboratory conditions are very
often of this nature. The fitted curve expresses a general trend, but
individual cases may lie well away from it in a number of instances.

Fitting of more general curves

15.20 Example 15.4 — We must now consider the fitting of curves of order
higher than the first.

Table 15.5 on p. 356 shows the percentage loss of weight (Y) for certain
temperatures {X) in experiments on the oven-drying of soils. Since X is
here the controllable factor, it is natural to take it as the independent
variable, and we shall express Y in terms of X.

The data are shown graphically in fig. 15.5. We shall find successively
the straight line, quadratic parabola and cubic ^parabola of closest fit. We
shall therefore require sums of powers of X up to and sums of

products up to S{YZ^). We also require, for later work, S(Y^).

The preliminary work is shown in Table 15.5. We might, perhaps,
have abbreviated the arithmetic slightly by taking an origin of x at
-X'=100 and of y at Y=3, but the saving would not have been large.
Data of this kind frequently give rise to large figures in the higher sums,
and a machine is a great help in the calculation. For instance, with a
machine the sums S(YZ), etc., can be found by continuous addition,
without the necessity for vmting each individual contribution in the
relative column.

For the straight line of closest fit, equations (15.8) become —

82 • 97 - 1 6^0 -264203^ =0
14,736- 19-2642(350 -474,050% =0

whence

(io=0*660 and %=0*02741
(more accurately, 0 • 659,759,789 and 0 • 027,408,722)

SIMPLE CURVE FITTING

353

TABLE 15.4. — ^Reduction of non»linear relation to linear form

Relationship between Working Costs per Unit and Number of Units Sold in 27 Electriaty

Undertakings

(Data from Return of Engineering and Fmancial Statistics, 1933-34 — Electriaty Commission.)

Name of
undertaking

Working
costs per
umt sold
(pence)

Units sold
(excluding
bulk
supplies)
per head of
population

I’ll

logl

Aberdare .

1-53

63-1

0-18469

1-8000

0-3324

3-2400

Barry U.D.C.

2*36

12 1

0-37291

1-0828

0-4038

M725

Bredbury and
Ronnley

0-70

394-2

-0-15490

2 5957

-0-4021

6-7377

Chesterfield

0*56

220-5

-0*25181

2-3434

-0-5901

5-4915

Earby

1-41

52-4

0-14922

1-7193

0-2566

2-9560

Grange

1-88

119-4

0-27416

2-0770

0-5694

4-3139

Holmfirth .

181-6

0-06819

2*2591

0-1541

5-1035

Lincoln

0-78

293-8

-0-10791

2-4681

-0-2663

6-0915

Mexborough

170 4

0-05308

2-2315

0-1185

4-9796

Nuneaton .

0 86

184-1

-0-06550

2-2651

-0-1484

5-1307

Redcar

1-91

68 0

0-28103

1-8325

0-5150

3-3581

Slaithwaite

1-40

80-7

0-14613

1-9069

0 2787

3-6363

Tanfield

2 41

29-0

0-38202

1 4624

0 5587

2-1386

WestLancsR D.C.

•1*37

53-4

0-13672

1 7275

0-2362

2-9843

Dumfries Corp.

93 0

0-04139

1-9685

0 0815

3-8750

Tobermory

4-21

19-9

0-62428

1-2989

0-8109

1-6871

Aberayron . '

8*9

25 6

0-94939

1-4082

1-3369

1-9830

Brixham Gas and
Electric Co.

3-13

30-4

0-49554

1-4829

0-7348

2-1990

Chudleigh Co.

7-28

16 7

0-86213

1-2227

1-0541

1 -4950

Foots Cray Co

1-92

77-8

0-28330

1-8910

0-5357

3-5759

Lewes Co

120-1

0-05690

2-0795

0-1183

4-3243

Newcastle Electric
Light Co

0*64

68-8

-0 19382

1-8376

-0-3562

3-3768

Ramsgate Co

1-57

I 60-5

1 0-19590

1-7818

1 0-3490

3-1748

Steyning Co

1-06

93-9

0-02531

1 9727

0-0499

3-8915

West Devon Co. .

1-98

22-1

! 0-29667

1 3444

0-3988

i 1-8074

Coatbridge and
Airdne Co

0*68

196-2

-0-16749

2-2927

-0-3840

5-2565

Skelmorhe Co

2 05

60-1

j 0-31175

j 1-7789

0-5546

3-1645

Total

—

i 5-24928

50-1311

7-3008

97-1450

and the straight line is —

=0-660 +0-0274 la: (a)

For the quadratic parabola, equations (15.8) are —

S(Y) -nuo -ai2(X) -a2S(X2)=:0

S(YX) -flj2(X8)=0

S{YX2) -aiS(A'3) -ag2:(X')=0

354

THEORY OF STATISTICS

LogoTjikui of nnmber of units sold per head of population.

Fig. 15.4. — Straight line fitted to logarithms of data of Table 15.4

These become, on substitution,

82 • 97 — 16<^o — 2642(^1 — 474,050^2 =0
14,736 • 19-2642^0 --474,050^^ -91 ,244,582^3 =0
2,81 9,909 • 45 -474,050^0 -91 ,244,582^1 - 18,553, 1 64,842^2 =0
giving

^0 =3 • 551 , = -0 • 009291 , 00010695

(more accurately, 3*550,990,2, —0-009,291,235,7, and 0-000,106,954,12)
and the parabola is —

y =3 • 551 -0 • 009291 a;+ 0 • 00010695:^«
For the cubic parabola, equations (15.8) are —

s(y)

S(yz)

S(yx==)

S(yx®)

-na^ — %S(X)

-a^HiX) ^a^Z{X^) -a^{X^)
-a^Z{X^‘) ~a^{X^) -a^{X^)
-a^Z{X^) -a^iX^) ~a^Z[X^)

-aiZ{X^)^=0

-a3S(Z*)=0

-a3S{Z®)=0

(5)

which become-

82-97 - 16a|,-264ai-474.050fla-91,244,582aa=0
14,736 • 19 - 2642a8 -474,0S0ai - 91 ,244,582a j- 18,553, 164,842a, = 0
2,819,909-45-474,050ao-91, 244.582a. -18.553,164,842a, -3,930,294,225,302a,=0
571, 902, 362-11-91, 244,582a, -18,553, 164, 842a, -3,930,294,225,302a, -858,077,668,755.250a, »=0

SIMPLE CURVE PITTING

355

It is not really necessary to write out the large numbers of the later
equations as fully as we have done, and a certain amount of approximation
is allowable. The student should, however, be careful not to introduce it
too soon, as neglected quantities may become of cumulative importance
in the solution of the equations.

By straightforward but rather strenuous arithmetic we find —

ao==7*783, 08940

=0*0005875, ^^ 3 = -0 • 0000009189

(more accurately,

ao=7 • 782,526,861 , -0 • 089,402,395,60

a2=0-000, 587, 479, 234,2, ^ 3 =-0-000, 000, 91 8 , 89 1,069, 8 )

The smallness of the coefficients and does not mean that they are
of minor importance, since in the equation for y they are multiplied by
terms in and which may be large.

The cubic parabola is, then.

y =7 • 783 -0 • 08940a; +0 • 0005875a;2 -0 • 00000091
which we may also write as —

V=7-783 -8-940,4;-5-875f -LV-O-Smef ~Y

100 \mj \mj

ic)

Fig. 15.5 shows the data graphically, with the straight line and cubic
parabola of closest fit.

Temperccture (degrees)

Fig. 15.5.'— Straight line and cubic parabola of closest fit to the data of Tible 15.5*

TABLE 1BS» Curve-fittiiig to exptess the relationship between temperatiiFe and percentage loss in weight of certain soli samples

(Data from J. R. Coutts, ** * Single Value * Soil Properties : V. On the Changes Produced in a Soil by Oven-drying/’ Journal AgncuUural Science, 1930, 20, 541.)

356

THEORY OF STATISTICS

SIMPLE CURVE FITTING

357

15.21 Although a graph will usually suggest whether a straight line
or quadratic parabola is likely to give a satisfactory fit, it will not as a rule
be much guide in deciding whether further terms will repay the labour
of calculation. This can be judged, at least roughly, by calculating
the terms given by the polynomial (to as high a degree as it has been
carried) for the observed values of a;, and then observing the run of the
residuals. If the signs run more or less at random it will hardly be
worth while to calculate another term , but if a series of positive residuals
is followed by a series of negative residuals, these by another series of
positive residuals, etc., it will probably be worth while to proceed further.
Moreover, the coefficients for a parabola of order k are no guide to those
of order ^+1- For instance, in Example 15.4, the values of for the
straight line, square parabola and cubic parabola are 0-660, 3-551, 7-783 ;
and those of are 0*02741, —0-009291, —0*08940. From this informa-
tion we could not guess even the sign of these coefficients in the parabola
of order 4, and if we wished to fit such a curve five equations of the type
(15.8) would have to be solved ah initio.

The student, therefore, should not fall into the error of thinking that
parabolas of successive orders will resemble each other in their lower
terms, or that the fitting of a curve of order ^ + 1 is merely a question of
adding an extra term to a curve of order k. It would be a great con-
venience if this were so, and, in fact, methods have been devised whereby
one variate can be expressed in terms of certain polynomials of the other
in such a way that this advantage is secured. The theory of these
so-called '' orthogonal polynomials is, however, outside the scope of
the present work.

The case when the independent variable proceeds by equal steps
15.22 When the independent variable x proceeds by steps of equal
amount h, the arithmetical solution of equations (15.8) can be greatly
simplified, particularly if the number of values is odd. In such a case
we take h as the unit of a: and an origin at the middle term. The values
of X will then be —k, — (^— 1), —(^—2), . . . —2, —1, 0, 1, 2, . . .
(^—2), (^— 1), k, and owing to the symmetry of this series the sums of
odd powers of x will vanish, i.e. S(ir), S(X®), S(A®), etc. are all zero.
Equations (15.8) then become, taking^ as odd,

S(Y) -^22(A2) . .

'L{YX) -aj:{X^) -^3S(^^) . . .

^YXP-^)-a^Z{XP-^) ^aoS(Z^+i) . . .
i:{YXP) ^aj:(XP+^) -^3S(Zi^+3) , ,

::i

(15.12)

and not only is the number of terms reduced, but the equations split
into two sets, one in a^, a,, ai, etc., and the other m a,, a^, a^, etc. More-

358

THEORY OF STATISTICS

over, the sums of even powers of X are twice the sums of powers of the
first k natural numbers, which may be easily found, either from tables
or from known formulae.

Example- 15.5. — Table 15,6 shows the population of England and
Wales in certain census years from 1811 onwards. Taking the time as
the independent variable, we choose as the unit of X the period of ten years,
and the origin at the mid-point of the range, 1871. The preliminary work
for the fitting of curves up to the cubic form is shown in the table.

For the cubic parabola, equations (15.8) are, then,

314-09-13<2o -182^r2

474-77 -182^21 -4550^23

4520 - 45 - 182^10 ~45S0a^

11,632-97 -4550^1 -134,342^3

whence

(20=23-299 (2i= 2-895

(22= 0-06153 (23=-0-01147

The parabola is, therefore,

y =23 • 299 +2 • 895:^ +0 • 061 53^:2 -O • 01 U7x^ (a)

Fig. 15.6 shows the data graphically, together with this cubic.

Incidentally, this example illustrates one point of some importance.
Over the years 1811 to 1931 the cubic gives a fair fit, and might be used
to estimate the population at intermediate years. But for extrapolation
it is of very little value. We could not estimate the population for 1961
with any confidence by putting in the cubic ; still less that for later
years. Unless there are good reasons for supposing that the fitted curve
is an accurate representation of a theoretical relationship, it is dangerous
to assume that a fitted parabola can be used outside the range for which
it was ascertained.

It would be instructive for the student to fit merely a segment of some
actual series and note how rapidly the curve calculated from the segment
diverged from the observations outside its limits. It has been shown that
even within the limits of the fitted observations the fit tends to be worst
as the limits are approached. The higher powers of x become of greater
and greater effect the more we diverge from the centre of the fitted
segment and tend, so to speak, to wag the tail of the curve.

15,23 If the number of values of x is even, we have a choice of two
methods of procedure. We can take h as unit and the origin at one of
the two middle values ; or we can take as unit and origin midway
between the two central values. In the first case, the sums of odd powers
will no longer vanish, but they will nevertheless be easily calculable.

SIMPLE CURVE FITTING

359

TABLE 15.6, — Curve-fitting to growth of population in England and Wales

(Data from Registrar-General’s Statistical Review of England and Wales, 1933, Tables, Part II.)

Year

Popu-

lation

(mil*ns)

A'®

yx*

1811

10-16

-216

1,296

46,656

-60*96

365-76

-2,194-56

1821

12-00

-5

-125

625

15,625

-60-00

300-00

- 1,500-00

1831

13-90

-4

- 64

256

4,096

-55*60

222*40

- 889-60

1841

15-91

- 27

729

-47-73

143-19

- 429-57

1851

17-93

-2

- 8

-35-86

71-72

- 143-44

1861

20-07

-1

- 1

-20-07

20-07

- 20-07

1871

22-71

—

— ^

—

1881

25-97

1891 1

29-00 !

58-00

116-00

232-00

1901

32-53

729

97-59

292-77

878-31

1911

36-07

256

4,096

144-28

577*12

2,308*48

1921

37*89

125

625

15,625

189-45

947*25

4,736-25

1931

39*95

216

1,296

46,656

239-70

1,438-20

8,629-20

Total

314-09

182

4,550

134,342

474-77

4,520-45

11,632*97

since all terms except a single otitl 5 dng member in the summation will
cancel out in pairs. In the second case the sums of odd powers will
vanish, but the other sums will no longer be twice those of the first k
natural numbers, but of the first k odd numbers. In either case the solution
of the equations (15.8) is not difficult.

36 o

THEORY OF STATISTICS

Calculation of the sum of squares of residuals

15.24 The eye is not a reliable guide to the closeness with which a given
curve lies to data, and it is desirable to have some more accurate measure
of the closeness of fit. For this purpose we require to be able to find
the sum of the squares of residuals U. We know by our method of
ascertaining the curve that this will be less than the corresponding quantity
for any other curve of the same degree, and our interest is centred on how
close this is to the ideal value zero.

To calculate the sum of squares of residuals it is not necessary to
calculate each separate residual. In fact, for the parabola of order p we
have —

V . . . ^opXP)^

. . . -^apXP)}

for the terms of the type . . . -—apXP)) vanish in

virtue of equations (15.8). Hence,

c 7 =s(y 2 )~ao 2 (y)-~ai 5 :(yx)-~ . . . -a^S(yz^) . (15.13)

The constants a and the sums which appear in this expression have
already been found, with the exception of Yi{Y^) in some cases. With
this additional quantity we can find U.

Example 15.6. — Let us find U for the data of Example 15.4 for the
straight line and the two parabolas.

For the line

[/-^(y 2) >-ao2(y) -a^{YX)

Here

S(y) =82 • 97, S(yz) =14,736 • 19

s ( y 2) =459 • 4363, =0 • 659,759,789

Hence,

^1=0* 027,408,722

U =459,4363 -54 • 74027 -403 -90014

=0-7959

For the quadratic parabola —

and here

whence

C;=S(y2) -a^{Y) -aiS(yZ) -a^scyz^)

^1= 3-550,990,2
^1^== -0.009,291,235,7
^2= 0-000,106,954,12

t/ =0-1271

Similarly, for the cubic

C7=0-0485

SIMPLE CURVE FITTING

361

The value of U therefore decreases from 0*7959 for the straight line to
0*0485 for the cubic. This is what we should expect, for the addition of
extra terms means that we have additional constants at our disposal in
the task of minimising U,

To obtain U with any accuracy by the foregoing method it is necessary
to ascertain the a's to a considerable number of decimal places.

Meastirement of the closeness of fit

15.25 The value of U enables us to make some sort of comparison
between the fits of different curves to the same data ; but it is not, in itself,
a satisfactory measure of fit, since it does not permit of the comparison
of the fits of curves to different data. The measure U In, which is the
variance of errors of estimation, suggests itself, but this, like U, is not
absolute, being dependent on the units in which we are working. For a
satisfactory measure some form of ratio would have to be taken.

Such a ratio arises in a natural way if we consider the correlation
between the actual values of Y and those predicted by the pol 5 momial,
L,et us, without loss of generality, suppose that the values are measured
from their mean, and let be the value given by the polynomial and
be the actual value. Then, as in 15.24,

2:(y2)=E(ry) . . . (15.14)

C7=E{y(y-y)}

=:S(y2)~S(yy) . . . (15.15)

Writing a^, (jy for the standard deviations of Y and y, and R for the
correlation between them, we get, from (15.14),

<Jy=^R<Jy .... (15.16)

and from (15.15),

^ 2 p

.... (15.17)

CTj.

Hence, substituting for cry from (15.16),

R^=l — (15.18)

which gives the correlation in terms of the ratio oil] jn and the variance

R is, in fact, analogous to the multiple correlation coefficient and the
correlation ratio, and the equation (15.18) should be compared with
equation (11.3), page 256, and equation (12.15), page 298.

362

THEORY OF STATISTICS

Example 15.7. — In Example 15.1 we have, using the data of Table 15.2
and the constants found —

a/-67‘28998-~(6- 126)2
=:29* 762,104

[7=1-835,777,255

1-835,777,255

297*62104

0*993,831,830

=0*99691

For the soil data of Examples 15.4 and 15.6 we find —

For the straight line =0*98627
For the cubic =0*99917

Thus, judged by the value of R, the straight line of Example 15.1 is a
better fit than that of Example 15.4, but a worse fit than the cubic of the
latter.

15.26 As a general comment on the scope of the methods of curve-
fitting described in this chapter, we may remark that although polynomials
can always be fitted to data, the student should not assume that even the
polynomial of closest fit will necessarily be a satisfactory fit. It may
exhibit peculiarities of behaviour which are entirely absent from the data
themselves. He may well ask, when confronted by a given set of data,
how he is to know whether they may be satisfactorily represented by a
polynomial. The answer is that he must fit one and see. Some further
remarks on this point are given later in 24.12, where similar questions
arise in connection with interpolation and graduation.

15.27 The reader must be mindful of the fact that in the type of curve-
fitting discussed above there is an essential difference between the roles
of the independent and the dependent variables, which accounts for
there being two curves according to which variable is regarded as in-
dependent. If y is the dependent and x the independent variable the
minimisation of the sum of squares of residuals in the manner of 15.8 is
equivalent to supposing that if there is a '' true '' law under which y is
equal to a polynomial in a;, the errors " observed are in the dependent
variable y, not in x. Per contra, if we suppose that the errors are in x,
we must minimise the sum of squares of residuals in x, which makes the
latter the dependent variable.

15.28 Suppose, however, that x andy are known to be related by a linear
equation but that both variables are subject to error. What is then the
appropriate method of finding the best estimate of the unknown relation ?
If the errors are small, as seems to be the case in Example 15.1 , an approxi-
mation is given by the methods we have used because the two lines of
closest fit are nearly identical. But where the errors may be large, and in
any case as a theoretical problem where both variates are subject to error,

SIMPLE CURVE FITTING

363

we may require to find a unique relation most probably (in some sense)
representing the truth* This sort of problem may very well arise, for
example, in physics where it is assumed that there exists a definite func-
tional relationship between two quantities (the pressure and the reciprocal
of the volume of a gas or the length and temperature of a metal rod) both
of which are subject to errors of measurement.

15.29 This type of problem is extraordinarily difficult to solve and we
have no space to discuss it here at any length. A single illustration of
the complications which arise will have to suffice.

A plausible procedure to determine a unique straight line fitting a set
of points on a scatter diagram is to minimise the sum of squares of per-
pendiculars from the points on to the line. This is equivalent to finding
the principal axis (10.9) which, in a sense, may be regarded as '' closest
to the points. But unfortunately this line will vary according to the
scale of measurement of the vanates — if we double the scale of one and
hence enlarge the scatter diagram by the factor 2 in one direction, the
new line has a different equation from the old and the difference is not
merely that the transformed vanate is in the new scale. Geometrically,
we may say that right-angles are not preserved in a diagram if it is
stretched in one direction, so that perpendiculars from points to lines
do not remain perpendiculars under such a transformation. The procedure
we are considering, therefore, whatever its merits as providing empirically
a line of closest fit, is open to the theoretical objection that the answer
it gives depends on the scale of measurement, which in many problems
is repugnant to commonsense requirements. We do not, for example,
expect the linear law connecting the length of a rod with its temperature
to depend on whether we are measuring the latter in Centigrade, Fahrenheit
or absolute units. The procedure is reasonably plausible if both variables
are of the same kind, e.g. both temperatures, so that a change of scale
affects both to the same extent. The difficulties become intensified if
the underlying law is not linear.*

SUMMARY

1. A parabola of the form may be

fitted to data by choosing the constants a so that the sum of squares of
residuals U . . . —apXP)^ is a minimum.

2. This method leads to the equations

S(y) -<h^{X) ^ . -apj:{XP) =0

S(YZ) -aoS(Z) -«iB(Z2) =0

i:{YXP) -a^I^iXP) ^a^{XP^^) ^a^{XP^^) - . . . -apZ[X^P) =0

♦ For a useful review of the problem see D.V. Lindley, Supp, /. Roy. Statist. Soc.,
1947 , 9 . 218 .

364

THEORY OF STATISTICS

3. Non-linear data may sometimes be reduced to the linear form by a
simple transformation of one or both the variables.

4. The sum of squares of residuals may be found from the formula

C/=i:(Y2)_a^S(y)_a^S(yX)- . . . ~apZ{YXP)

5. One measure of the goodness of fit of the parabola to the data is
given by R, the correlation between actual and “ predicted ” values of the
variate. R is given by

where Y is the dependent variable.

EXERCISES

15.1 Fit a straight line and parabolas of the second and third orders to
the following data, taking X to be the independent variable —

X Y

0 1

1 1-8

2 1-3

3 2-5

4 6-3

and find the sum of squares of residuals in the three cases.

15.2 (Data quoted by P. L. Fegiz, “ Le variazioni stagionali della
natality,” Meiron, vol. 5, 1925, No. 4, p. 127.) The following figures
show the relation between duration of marriage and average number of
children per marriage in Norway in 1920 —

Duration of marriage
(years)

0 - 1
5- 6
10-11
15-16
20-21
25-26
30-31

Average number of
children
0-48

2- 09

3- 26

4- 33

5- 14
5-63
5-77

By the method of least squares find equations of the first, second and third
orders expressing the number of children in terms of the duration of
marriage. Compare the values given by these expressions for a duration
of 17-18 years with the true value 4*67.

SIMPLE CURVE FITTING

365

15.3 The pressure of a gas and its volume are known to be related by an
equation of the form pvy —constant.

In a certain expenment the following volumes of a quantity of the
gas were observed for the pressures specified. Find the value of 7 by

fitting a straight line to the logarithms of p and v, taking p to be the
independent variable.

p (kg. per square cm.) . 0*5

t; (litres) . . . 1*62

1-0 1-5 2-0 2-5 3-0

1-00 0-75 0-62 0-52 0-46

15.4 The following are the gross output and the gross output per £100
of labour employed, for a selected number of farms —

Gross output
(units)

Gross output
per £100 labour
(units)

223

155

755

188

165

1,535

315

3,193

290

2,238

259

1,228

231

2,695

255

Fit a quadratic parabola to these data, taking gross output as the in-
dependent variable.

CHAPTER SIXTEEN

PRELIMINARY NOTIONS ON SAMPLING

The problem

16.1 In practical problems the statistician is often confronted with
the necessity of discussing a population of which he cannot examine every
member. For example, an inquirer into the heights of the population
of Great Britain cannot afford the time or expense required to measure
the height of each individual ; nor can a farmer who wants to know what
proportion of his potato crop is diseased examine every single potato.

In such cases the best an investigator can do is to examine a limited
number of individuals and hope that they will tell him, with reasonable
trustworthiness, as much as he wants to know about the population from
which they come. We are thus led naturally to the question : what
can be said about a population when we can examine only a limited
number of its members ? This question is the origin of the Theory of
Sampling.

16.2 A sample from a population is a selected number of individuals
each of which is a member of the population. As a very special case the
sample may consist of the entire population.

It is a matter of common belief, founded on experience and intuition,
that a sample will tell us something about the parent population. The
com merchant, whose livelihood depends on his ability to ascertain
the quality of the grain which he handles, is content to assess it by thrust-
ing a conical trowel into the middle of a sack and scrutinising the sample
he gets. He believes that the sample will be representative of the whole,
and experience justifies him. He buys and sells on the basis of judgment
from samples. It is also a matter of common belief that the larger a
sample becomes the more likely it is to reflect accurately the conditions
in the parent population.

To these and similar beliefs the theory of sampling gives a logical
basis and a system of quantitative measurement. In this chapter we
give a general survey of the fundamental ideas and the technique of
sampling. In later chapters we shall develop these ideas and discuss their
applications in various fields.

Types of popidation

16*3 Before we consider sampling itself, however, it is desirable to look

366

PRELIMINARY NOTIONS ON SAMPLING 367

a little closer into the various types of population which we shall have
to investigate.

By a finite population we shall mean a population which contains a
finite number of members. Such, for instance, is the population of
inhabitants of Great Britain and the population of books in the British
Museum,

Similarly, by an infinite population we shall mean a population containing
an infinite number of members. Such, for instance, is the population of
pressures at various points in the atmosphere, or the population of
possible sizes of the wheat crop, for, although there are limits to the
size, the actual tonnage can take any numerical value within those limits.

In many cases the number of members in a population is so large as to
be practically infinite. Moreover, a theoretical discussion of an infinite
population is frequently easier than a discussion of a finite population, and
a large class of problems may be treated by assuming that the parent
population is infinite, without introducing any sensible error.

It may be worth remarking that in a few cases we may be ignorant
whether or not the population under discussion is infinite. The population
of stars is an example.

Existent and hypothetical population

16.4 By the logical extension of the idea of a population of concrete
objects, which we shall call an existent population, we are able to construct
the idea of a hypothetical population.

Consider the throws of a die. Each throw will be regarded as an
individual. There is an infinite number of throws which can be made
with the die, provided that it does not wear out. Let us then define as
our population of discussion all the possible throws of the die.

In doing so we are clearly making some new step ; for our population
is to be conceived as having no existence in reality but only in imagination.
We can give actuality to some members of the population by throwing the
die, but we can never produce them all. Even if the die were locked
away in a safe and never thrown at all there would still be a population
of possible throws.

Such a population is called a hypothetical population. We may define
it formally as the aggregate of all the conceivable ways in which a specified
event can happen. Other examples of hypothetical populations are the
population of all values which the bank rate can have in ten years' time,
and the population of the possible ways in which three balls can be
arranged on a billiard table.

16.5 A hypothetical population may, in fact, be imagined around
any observed event. We have only to picture all the circumstances
before the event happens ; the population is then all the possible ways in
which it could happen. Which of the ways it will happen does not affect
the population. We know that *'from the chaos of predestination and

368

THEORY OF STATISTICS

the night of our forebeing some one individual will emerge to assume
the mantle of reality ; but which one that will be is another and more
difficult question.

16.6 The student of metaphysics would perhaps criticise the thoughts
expressed briefly in the previous two sections, but we have no space to
go further into the philosophical implications of the idea of hypothetical
populations. The problems which arise in this connection have, however,
far more than an abstract interest. They lie at the root of a great many
practical statistical problems, and most students, however utilitarian
their outlook, will find that a clear perception of the issues involved may
save a lot of thought and labour at a subsequent stage.

Population of populations

16.7 Just as a population may contain a number of sub-populations,
so any given population may be a member of some more widely defined
population. For example, the population of inhabitants of Great Britain
is a member of the population of populations, each of which consists of
the inhabitants of some European country.

Similarly, any existent population may be regarded as one member of a
hypothetical population of populations. For instance, the normal popula-
tion of men whose heights have a mean of 65 inches and standard
deviation 3 inches is a member of the hypothetical population of all
populations which are normally distributed with respect to height.

16.8 We shall sometimes have to discuss aggregates which it is difficult
to regard as composed of individual members at ail — for example, we
may wish to sample a reservoir of water to test for pollution. In theory,
perhaps, we could m such a case regard the reservoir as a population
composed of molecules each of which was an individual, but in practice,
as we shall see, this is not usually a convenient method of approach.
Such populations may frequently be treated as composed of arbitrary units,
e.g. the reservoir may be regarded as composed of so many pints of fluid.
Similarly, a 280-lb. sack of flour may be regarded as composed of 4,480
omices, and we can, if we like, regard it as weighed out into one-ounce
packets.

16.9 We can now turn to discuss the aims which usually underlie a
sampling inquiry.

Briefly, fhe fundamental object of sampling is to give the maximum
information about the parent population with the minimum effort. We
must, therefore, consider the type of information we require and the
methods by which it is to be obtained.

16.10 In sampling a population we usually have in mind one or more
of its variates. For instance, when we sample the population of Great
Britain, we are not so much interested in the individuals as human beings
as in one of their qualities, such as height or weight, or perhaps the correla-

PRELIMINARY NOTIONS ON SAMPLING

3^9

tion between height and weight Our object will then be to get, from the
sample, an idea of the frequency-distribution in the parent population
according to the chosen variates.

The ideal for the purpose would be to express this distribution in some
mathematical form such as a Pearson curve (8.48). It may be, however,
that the parent population will not admit of this representation, or that the
sample is not large enough for us to venture on it with any confidence.

In such cases we attempt to find estimates of certain constants of the
parent population. Very often this is all we need. We can, for example,
form a very fair idea of the height distribution of the population of Great
Britain if we know the mean and the standard deviation. If we can go
further, and find the third and fourth moments, our idea will be better still.
Theory of estimation

16.11 Hence, a large part of the theory of sampling is devoted to finding
from the sample estimates of certain constants of the parent population.
Such constants include the measures of position and of dispersion together
with the moments and measures of skewness ; and, in multivariate
populations, the various total and partial correlations.

In general, there are more ways than one of estimating a constant from
the data of the sample. Some of these ways will be better than others.
The Theory of Estimcition treats of these and cognate matters. It seeks
to investigate the conditions which an estimate should obey, what are
the best estimates to employ in given circumstances, and how good other
estimates are in comparison.

Precision of estimates

16.12 It will be obvious that knowledge derived from a sample is not
of the categorical kind customary in mathematics. If we have 1 ,000 balls
in a bag and draw 999 of them which turn out to be black, it is always
possible that the remaining one is of some other colour. It is, however,
so improbable, that in most practical cases we should be justified in con-
cluding that the balls were all black.

If we did draw such a conclusion, and acted upon it, we should be basing
our action, not upon certainty, but on probability. One does this kind
of thing, of course, in nearly all everyday actions almost without noticing
it. Some events, such as the death of a man before reaching the age of
150, have such a high degree of probability that we never regard them as
other than certain ; other events, such as the possibility of rain to-morrow,
are so uncertain that we should hesitate to make an important decision
contingent upon them.

16.13 The second aim of the theory of sampling is, therefore, to determine
as objectively as possible what degree of confidence we can put in our
estimates when they are obtained. This we do in terms of probability
as far as we can ; if this proves impossible, we sometimes have to rely
on intuitive impressions or the results of previous experience, which
are not expressible in quantitative terms.

370

tttEORY OF STATISTICS

Put in another way, we may say that our object is to determine the
precision of an estimate. We attempt to do this by assigning limits to
the probable divergence between the estimate based on the sample and the
true value of the estimated quantity in the population.

16.14 The accuracy of the estimate will depend on (a) the way in which
the estimate is made from the data of the sample, and (6) the way in
which the sample was obtained. Consideration of the first leads us
again to the theory of estimation. The second leads us to study the
technique of sampling and the design of statistical inquiries.

Tests of significance

16.15 If the sample is small we cannot, as a rule, assign to the estimates
we obtain sufficiently narrow limits to locate the population value with
any serviceable accuracy. For example, a correlation of +0*5 in a
sample of twelve might arise, rather infrequently, from a normal popula-
tion in which the true correlation was as high as +0*9 or as low as zero.
For such samples our questions are accordingly framed in more qualitative
terms : we do not ask, “ What is the value of the correlation in the
population ? '' but, '' Is the observed value significant of the existence of
any correlation at all in the population, whatever its value ? '' In other
words, we wish to know whether the observed value could have arisen
from a population in which the true correlation is zero. If our conclusion
is that it could not, we may say that the sample value is significant of
correlation, although we cannot say with much confidence what that
correlation is.

Much of the investigation arising out of small samples is thus of a rather
special character, and deals with tests of significance. The methods
developed for the purpose of conducting such tests can be, and not in-
frequently are, applied also to large samples, either alone or supplementary
to the direct approach of forming more or less precise estimates of the
various quantities which specify the parent population.

Types of sampling

16.16 The process of forming a sample consists of choosing a predeter-
mined number of individuals from the parent population The choice
may be exercised in three ways —

{a) By selecting the individuals at random (the meaning of “ random "
is discussed below).

(6) By selecting the individuals according to some purposive principle.

Thus, in taking a sample of the inhabitants of Great Britain to study
their income w^e might, according to method {a), select the individuals
at random from census returns ; or according to {b) we might, knowing
toughly the average incomes in various age-groups, purposely select from
each group an individual whose income was somewhere near the average

PRELIMINARY NOTIONS ON SAMPLING

371

in that group ; or (c) we might decide to take ten individuals from each
group and select those ten by method (a),

16.17 Sampling of type (a) is called random sampling. That of type
{h) is called purposive sampling. That of type (c) is sometimes referred
to as mixed sampling. If the population is divided into '' strata '' by
purposive methods and then a portion of the sample is taken from each

stratum,” the sampling is said to be stratified.

The application of each of these types may be affected by what is kno\m
as bias. This is the name given to perturbations which influence the
nature of the choice and make it something other than what the experi-
menter intends it to be. Bias may be due to imperfect instruments, the
personal qualities of the observer, defective technique, or other causes.
Like experimental error, it is difficult to eliminate entirely, but usually
may be reduced to relatively small dimensions by taking proper care.

By an obvious extension of the nomenclature, we talk of a sample
obtained by random sampling as a random sample, that obtained by
purposive sampling as a purposive sample, and so on.

Random sampling

16.18 The reader no doubt already has some intuitive ideas about
randomness of choice. We may give a formal definition of random
sampling by saying that the selection of an individual from a population is
random when each member of the population has the same chance of being
chosen. Similarly, a sample of n individuals is random when it is chosen
in such a way that, when the choice is made, all possible samples of n have
an equal chance of being selected.

16.19 The first question arising out of this definition which we have
to consider is : How are we to obtain a random sample ^

This question is more difficult than it appears at first sight. It might
be thought that any purely haphazard method of selection would give a
random sample. For example, if we wished to obtain a random sample of
local tradesmen, one way which suggests itself is to take a Trades Directory,
open it at random ” and take the first name on which the eye alights,
repeating the process until the sample is of the required size. Or again,
if we wished to obtain a random sample of wheat growing in a field, it might
be thought that a satisfactory method would be to throw a hoop in the air

at random ” and select all the plants over which it fell.

16.20 That such methods are apt to be deceptive may be seen from
the two examples we have j ust given. In the first, if we consulted a Trad^
Directory which had already been used, we should probably find that it
opened at some pages more readily than at others ; we should therefore
tend to get the more popular tradesmen. Moreover, our eye might tend
to be caught by long names or peculiar names. In either case some trades-
men would have a greater chance of being chosen than others, and the
sample would not be random.

372 THEORY OF STATISTICS

TABLE 16.1. — Height measurements of wheat. Frequencies of plants chosen by

eye in ranks 1-8

F. Yates, “ Some Examples of Biased Sampling,” Annals of Eugentcs, 1935, 6, 202

'Rtmks

Q>)

Fig, 16.1. — Distribution of wheat plants according to height (Table 16.1)

(a) Distribution of shoot heights (31st May) in ranks 1-8

(b) Distnbutionbf ear heights (28th June) in ranks 1-8

PRELIMINARY NOTIONS ON SAMPLING

373

Again, in the second example, our hoop might tend to be caught by the
taller ears of wheat, or we might tend unconsciously to throw it towards
parts of the field where the wheat looked to be about the average height.
These and other factors would destroy the random character of the
sampling.

Human bias

16.21 Experience has, in fact, shown that the human being is an
extremely poor instrument for the conduct of a random selection. Wher-
ever there is any scope for personal choice or judgment on the part of the
observer, bias is almost certain to creep in. Nor is this a quality which
can be removed by conscious effort or training. Nearly every human being
has, as part of his psychological make-up, a tendency away from true
randomness in his choices.

We may illustrate the unreliability of free choice on the part of even a
trained observer by taking an example of height measurements in samples
of wheat plants. In the course of certain work at the Rothamsted
Experimental Station, sets of eight wheat plants were selected for measure-
ment. Six of these shoots were chosen by purely random methods. The
other two were chosen '' at random '' by eye. If, in any set, the eight
shoots were ranged in order of magnitude, the two chosen by eye could
have any places from one to eight ; and if they, in common with the other
six, were really random, they should have occupied these places with equal
frequency in a reasonably large number of sets. Table 16.1 shows the
resulting frequencies in the ranks one to eight for 116 sets taken on
31st May (before the ears of wheat had formed) and 112 sets taken on
28th June (after the ears had formed).

Fig. 16,1 shows the same results graphically, the dotted line giving
the frequencies to be expected if the choice was really random.

The divergence of the actual from the expected results is very striking,
and clearly cannot be attributed to fluctuations of sampling. It will be
seen that on 31st May, before the ears had formed, the observer was
strongly biased towards the taller shoots ; whereas in June, after the
ears had formed, he was biased strongly towards a central position and
avoided short and tali plants.

16.22 Sight is not the only sense which may bias a sampling method.
In certain experiments counters of the same shape but of different colours
were put into a bag and chosen one at a time, the counter chosen being
put back and the bag thoroughly shaken before the next trial. On the
face of it this appears to be a purely random method of drawing the
counters. Nevertheless, there emerged a persistent bias against counters
of one particular colour. After careful investigation the only explanation
seemed to be that these particular counters were slightly more greasy
than the others, owing to peculiarities of the pigment, and hence slipped
thit>ugh the sampler's fingers.

374

THEORY OF STATISTICS

The student may perform similar experiments for himself. One of
the simplest is to ask a friend to recite “ at random one hundred digits,
including zero, and then count the number of odd ones. If the numbers
are really random, the number of even ones and odd ones should be about
equal, but there will frequently be found a bias one way or the other.

16.23 Enough has been said to show that if we are to evolve a satisfactory
method of random sampling we must eliminate all personal choice. The
method of selection must, therefore, follow some code of procedure which
leaves nothing to the observer's idiosyncrasies.

It may sound a little paradoxical to obtain true randomness by follow-
ing rules of procedure. We are reminded of Bertrand's question : " How
can we talk of the laws of chance, which is the negation of all law ? "
The ensuing sections will, it is hoped, remove any doubts on this head.

Technique of random sampling

16.24 The methods adopted in any given case to ensure as far as possible
that the sampling is random depend to some extent on the size and nature
of the population. Certain modes of procedure which are convenient
for small populations are not so for large populations. We shall also
see that sampling from a hypothetical population has a special significance
and special difficulties of its own.

16.25 The criterion that every individual should have an equal chance
of being chosen may be put in a somewhat different form. If the method
of selection is independent of the properties of the sampled population
which it is desired to investigate, there will, so far as those properties are
concerned, be no reason why one individual should be chosen rather than
another. Hence all values of the properties which occur in the population
will have an equal chance of being chosen. If, therefore, we can produce
a mode of procedure which bears no relation to the properties of the
parent population which we are discussing, we may expect that it will give
a random sample, so far as those properties are concerned.

16.26 We may now consider a few examples of the kind of procedure
to which this rule leads.

Suppose we wish to take a sample of the inhabitants of a street. They
are already arranged in houses, and for the sake of simplicity we will take
our problem to be that of selecting a number of houses, whose occupants
will comprise our sample.

Let us take as our rule of procedure the selection of every tenth house,
starting at some arbitrary point. Unless there are peculiar circumstances,
it is presumable that the properties we are investigating, which may,
for instance, be income or size of family, are not grouped periodically
along the street. The method of selection is then independent of the
properties of the population and the sampling will be random.

If, however, the street were divided into blocks by cross-streets at

PRELIMINARY NOTIONS ON SAMPLING

375

every tenth house, so that every house in our sample was a corner house,
and therefore, possibly, a shop, it is easy to see that the sample is no longer
random. Shops occur, in fact, along that street with period ten, and
since our method of selection has also that period, the method and the
qualities under investigation are no longer independent.

16.27. We might then fall back on a different method. If we take
a pack of plain cards, as similar as we can get them, we can make one card
correspond to one of the houses by writing on it the number of the house
in the street. The pack would then be a kind of miniature of the popula-
tion for sampling purposes. We can draw a sample of houses by drawing
a sample of cards, and if we shuffle the pack well we have every reason to
hope that a random sample will result, for it is hard to imagine any way
in which the method of shuffling and drawing could be dependent on the
properties of the population. It is not impossible to make it so, however.
For instance, if the ink with which we wrote the numbers on the cards was
slightly adhesive, the larger numbers would not be so easy to draw out
as the small ones, and we should tend to get houses at one end of the
street. If such houses were of the poorer class, our sample for the purpose
of investigating income would not be random.

Lottery sampling

16.28 The method we have just described, of constructing a miniature
population which is easily handled, is one of the most reliable methods
of drawing a random sample. It is the method usually adopted in drawing
the winning numbers in sweepstakes and lotteries. In such cases the
population is the aggregate of persons owning tickets in the lottery. To
every member of this population there corresponds a number, the totality
of which numbers, written on pieces of paper, comprises the miniature
population. In practice, these pieces are placed in similar containers,
usually small metal cylinders, and thrown into a large rotating drum, in
which they are thoroughly mixed or “ randomised.'"

16.29 The practical difficulties of constructing the miniature population
and of shuffling it are, however, severe if the parent population is at
aU large The method is, of course, inapplicable on theoretical grounds
if the population is not finite. To save the trouble of work with tickets it
is often possible to use numerical methods.

Suppose we require a set of points on the celestial sphere, as for example
if stars were uniformly distributed and we wanted a sample of stars. We
will take a point to be defined on the celestial sphere by latitude and longitude
(though this is not the way in which astronomers usually express it), and will
ignore difficulties arising from the existence of double stars or unresolved
objects. What we want, then, is a set of random pairs of latitudes and
longitudes. As a crude method we might take an atlas of the world and
choose the figure set out in the index for places arranged alphabetically.
But it is easy to see that this method is unsound ; for there will be more

376

THEORY OF STATISTICS

names associated with the more populous districts, and hence the values
given in the index will tend to cluster round certain points and avoid
others — there will be none in the middle of seas or at the poles, so that
the pole star has no chance of being selected.

Let us then take a set of statistical tables and open it haphazardly.
We shall be confronted with a page of figures, and if we take, say, the tenth
figure in each row we shall probably get a set of digits which are random.
Suppose the first ten digits obtained in this way were 7, 0, 4, 7, 9, 6, 8,
2, 9, 1. We might then take our star to be defined by latitude 70® 47*9'
and longitude 68® 29*1 . Another page will give us another star, and
so on.

Random sampling numbers

16.30 The difficulty in applying the method we have just described
lies in ensuring that the numbers we obtain are really random. Many
tables of figures, such as logarithm tables, may fail to give random digits
because there is a relation between the figures in successive rows. To
obviate this difficulty certain Tables of Random Sampling Numbers have
been constructed.

One such set, due to L. H. C. Tippett, con.sists of 41,600 digits taken
from census reports and combined by fours to give 10,400 four-figure
numbers. We give here the first forty sets as an illustration of their
general appearance —

2952 6641 3992 9792 7979 5911 3170 5624

4167 9524 1545 1396 7203 5356 1300 2693

2370 7483 3408 2762 3563 1089 6913 7691

0560 5246 1112 6107 6008 8126 4233 8776

2754 9143 1405 9025 7002 6111 8816 6446

The reader may wonder how it was ensured that these digits are random.
They were chosen haphazard, but the real guarantee of their randomness
lies in practical tests. We may say at once that Tippett's numbers have
been subjected to numerous investigations w^hich make their randomness
for many practical cases highly probable. A further set of numbers
(100,000 in all) was constructed by Kendall and Babington Smith using
a randomising machine. These kiso were carefully tested after con-
struction. The use of random sampling numbers will be apparent from
the following examples—

Example 16.1. — ^To take a random sample of 10 from the population of
8585 men of Table 4.7, page 82.

Here we have 8585 individuals. We will number them from 1 to 8585.
The problem of selecting ten men at random is then that of finding ten
numbers at random between 1 and 8585. We therefore take a page of
random sampling numbers and select the first ten on the page which are not
greater than 8585, Thus, if our page were the one on which appear the

PRELIMINARY NOTIONS ON SAMPLING 377

numbers we have quoted above, our individuals would be those correspond-
ing to the numbers, reading across.

2952, 6641, 3992, 7979, 5911, 3170, 5624, 4167, 1545, 1396

If we imagine the numbering to be done in order of height, starting with
the shortest and ending with the tallest, we see that the first individual falls
in the group 66—'', the second in the group 69—'', and so on. The height-
ranges in which the ten individuals fall are, in fact, in inches —

66-, 69-, 67-, 71 68-, 66-, 68-, 67-, 65-, 65-

Let us take their heights as being given by the centre points of these ranges,
and find their mean. We have—

. . +65)

= 67*2

Hence the mean is 67 *6 inches, as against the true value of 67*46 inches in
the whole population.

Example 16.2. — ^To take a sample of 5 from the distribution of screw
lengths of Table 4.3, page 72.

Here we have 206 individuals. It would clearly be a waste to use only
numbers from 0001 to 0206 for the screws and to neglect the rest, and we
are able to bring nearly all numbers into play by the following device.
We note that 206 goes 48 times into 10,000, with a certain remainder. In
fact, 206x48=9,888. We therefore attach 48 numbers to each screw.
Taking them in order, beginning at the shortest, we let the first screw
correspond to the numbers 0001 to 0048, the second to 0049 to 0096, the
third to Q097 to 0144, and so on, the 206th screw corresponding to the
nuifibers 9841 to 9888. Numbers above 9888 we leave out of account.
Referring to the table, we see that there is one screw in the first category
(5 to 6 thousandths short of an inch), four in the second (4 to 5 thousandths
short of an inch), and so on. The numbers corresponding to screws in the
different categories will then be 0001-0048, 0049-0240, 0241+1768, and
so on ; or, in tabular form.

We now take five random sampling numbers from the tables. For
instance, w'^e might take the five in the first column of 16.30, i.e. 2952,
4167, 2370, 0560, 2754. The screws corresponding to these numbers will
be 1 • 5, 0 • 5, 1 * 5, 3 • 5 and 1 • 5 thousandths short of the inch respectively.

If we had obtained two numbers, say 0001 and 0002 in the first category,
we should have been faced with the necessity for a decision on how the
sampling was to be regarded, for there is only one screw in this category.
If we suppose that a sampled screw is abstracted from the population, it can
only be drawn once ; and hence we should have had to ignore all numbers
in the category 0001 to 0048 subsequent to that which first occurs. If, on
the other hand, the screw is replaced, we can draw it as often as we like.

378

THEORY OF STAliSTICS

DiSereace in
length from

1 inch

(thousandths)

Numbers

corresponding

Difierence in
length from

1 inch

(thousandths)

Numbers

corresponding

— 6 to —5
~5 to —4
-4 to -3
-3 to -2
-2 to -1
-I to 0

0 to + 1

0001—0048
0049—0240
0241—0768
0769—1824
1825—3024
3025—4320
4321 -5856

-fl to +2
+ 2 to +3
+ 3 to +4
+4 to +5
+ 5 to +6

5857—7488

7489—8688

8689—9456

9457—9840

9841—9888

Example 16.3. — In Example 2.5, page 25, we had the following data
g’vmg the association between inoculation against cholera and exemption
ftom attack in 818 subjects —

Not attacked

Attacked

Total

Inoculated

276

279

(0001-3312)

(3313-3348)

Not inoculated .

473

539

(3349-9024)

(9025-9816)

Total

749

69 j

818

Let us take a sample of 10 from this population.

We observe that 818 goes into 10,000 twelve times, with a certain
remainder. In fact, 10,000=12x818+184. We can therefore attach
12 random sampling numbers to each member of the population. To the
276 inoculated-not-attacked individuals we attach the numbers 0001 to
3312 (12x276). To the 3 inoculated-at tacked individuals we attach the
numbers 3313 to 3348 (a range of 36, equal to 3x12). Similarly for the
remaining individuals. The random sampling numbers corresponding to
the individuals in the four compartments of the table are shown in brackets
above.

We then take ten random sampling numbers from the tables, say the
first ten, reading across, from the numbers given in 16.30. If we had
come across a number greater than 9816 we should have ignored it. The
first number, 2952, gives us an individual falling in the inoculated-not-
attacked class ; the second, 6641, gives us a member of the not-inoculated-

preliminary notions on sampling

379

not-attacked class ; and so on. The 10 numbers give the following
results —

Not attacked

Attacked

Total

Inoculated

Not inoculated .

Total

1 2

Example 16.4. — Strictly speaking, random sampling numbers are
applicable only to sampling from a finite population, for we cannot attach a
different number to each member of an infinite aggregate. But, by the
following device, we can apply the tables to draw samples from a con-
tinuous (and therefore infinite) population which is specified by a mathe-
matical equation in such a way as to give us the proportion of the total
frequency in given ranges of the variate.

In fact, let us draw a sample from a normal population with unit
standard deviation and unit total frequency.

Let us take ranges of 0* 1 on each side of the central ordinate. Table 2
of the Appendix will then give us the proportion of the frequency lying
in these ranges. As in Example 16.1, we divide up the numbers from
0000 to 9999 in pioportion to these frequencies, and this is, in fact, a par-
ticularly simple matter. All we have to do, for the positive values of the
variate, is to take the figures in the table, which have four figures. For
example, for the first interval 0*0 to 0*1, there will correspond the
numbers 5000 to 5398 ; to the interval 0*1 to 0*2, the numbers 5399
to 5793 ; to the interval 0*2 to 0*3, the numbers 5794 to 6179; and
so on. For the negative values of the variate we have, similarly, for 0*0
to ~0* 1, the numbers 4601 to 4999 ; for —0* 1 to —0*2, the numbers 4206
to 4600 ; for —0*2 to —0*3, the numbers 3820 to 4205 ; and so on, there
being as many numbers in any negative range as in the corresponding
positive range. Occasionally doubt may arise in assigning a number to a
given interval owing to the difficulty of rounding up a figure ending in 5.
In practice it is not likely to make any difference which interval we
choose ; if it threatens to do so, we can take the doubtful number to refer
alternately to the two possible intervals.

Having assigned numbers to the ranges, we select from the random
sampling numbers tables in the ordinary way. For instance, a number
5500 will correspond to a member in the range 0* 1 to 0*2. If we wish
to ascertain the mean of a sample, or some similar function of the variate
values, we take the variate value of any individual to be the centre of the
interval in which it falls. This is an approximation, but the narrowness
of the intervals justifies it in most practical cases.

* 8 ©

THEORY OF STATISTICS

Sampling from infinite populations

16.31 The methods we have just been discussing are appropriate only
to those cases in which the population is finite, so that it was possible
to associate with each individual one or more random sampling numbers ;
or to populations which, though infinite, can be treated by the method
of Example 16.4 owing to their complete specification according to the
variate under discussion. The required conditions are met with in much
of the material treated in practice, particularly in demographic and
economic work ; but m other work the population may be either infinite
or so large as to be infinite for all practical purposes, and a different
technique must therefore be used.

Consider, for example, the problem of drawing a random sample from
a sack of hour. We clearly cannot number all the particles in the sack,
nor could we extract any given particles and examine them. We might,
perhaps, reduce this case to that of a finite population by weighing out the
flour into small, say one-ounce, packets and then sampling the packets.
This is a kind of mixed sampling. But it is also possible to handle the
problem by a special technique, as follows.

First of all, we mix the flour thoroughly. We then divide it into
two halves and select one half. (It does not matter which, but for con-
venience we may imagine two heaps, one on the right and one on the left,
and select left and right alternately.) We then divide the half we have
chosen into two further halves, and again select one. The process is
continued until the sample has reached a manageable size. We may
reasonably suppose that it is random, especially if the flour is well mixed
at each stage before being divided into two.

A similar technique may be used for many “ continuous substances,
such as milk, grain, cement, etc.

Sampling from hypothetical populations

16.32 The technique for drawing random samples brings out a funda-
mental difference between existent and hypothetical populations. Taking
a simple but typical case, let us draw a sample from the population of
tlirows of a die.

The methods we have previously used are quite obviously inapplicable
here. We cannot construct a card population, because we do not know
the nature of the parent population. Nor can we put all the possible
throws in a heap, and select from it by continued subdivision. In fact,
there is only one thing we can do, and that is to throw the die, and take
our results as a sample.

What reason have we to suppose that this is a random sample ? The
answer lies partly in theory and partly in technique. In the first place,
we must adapt our method of throwing so that the sampling conditions,
so far as we can see, remain constant throughout the experiment. This
m a matter of technique, and our methods can, in fact, be tested* But

PRELIMINARY NOTIONS ON SAMPLING

381

Since our population does not exist for us to examine separately, the only
knowledge about it being derived from t}ie sample itself, it will be clear
on a httle reflection how difficult it is to say that every other possibility
in the population had an equal chance of occurring. We return to this
point in 16.35 and 16,36 below. Ba^caJlly our assumption is that our
throws behave as if they were being chosen at random from an existent
population. The justification for this is our general knowledge of the
behaviour of dice.

The importance of random sampling

16.33 We have already remarked on the importance of being able to
gauge the error of an estimate made from a sample. The practical use
of the theory of random sampling lies largely in the fact that it allows
us to measure objectively, in terms of probability, errors of estimation or
the significance of a result obtained from a random sample. The purposive
methods to which we refer below do not do this, or at least have not yet
been made to do so. The present trend among statisticians is, therefore,
on the whole, in favour of the use of random sampling methods except in
certain special cases.

16.34 At this point we may bring forward two important considerations.

In the first place, it must not be forgotten that random samphng may

produce the most unrandom-looking results. For instance, we usually
regard a hand of cards at bridge as a random sample from the population
of 52 which comprise the pack ; but it is not unknown for a hand of
13 spades to be dealt. The fact that the sample looks purposive, there-
fore, proves nothing. But it does provide a basis for strong presumptions.
How strong those presumptions may be the student may judge for himself
by imagining what he would think of a card party at which he got 13
spades twice in succession.

Secondly, we can never be absolutely certain that a method of sampling
is random. There are doubts on a priori grounds because for any given
method there are always conceivable sources of bias, and we can never
rule out entirely the possibility that some of these sources 'are present.
The utmost we can do is to make their presence extremely unlikely by
taking great care with the experiment.

16.35 We can, however, apply tests to judge the randomness of a
sampling method. If we draw a single sample from a known population,
the result will tell us nothing about the method adopted ; but if we take
a large number of samples they should, if the sampling is ’random, be
distributed in a certain way, and for some populations we can calculate
mathematically what that way ought to be. If, therefore, we apply our
sampling method to such a parent population and find the results widely
divergent from expectation, we have every reason to suspect our sampling
technique. Per contra, if the results and expectation are in accord, there
is good ground for reliance on the sampling.

3^2

THEORY OF STATISTICS

16.36 Tests of this kind presuppose that we know the form of the parent
population. In sampling from a hypothetical population we do not
know this, and are forced to estimate it from the sample. Clearly, we
cannot use this estimate to criticise the method by which the sample was
obtained without some closer inquiry.

Similar problems may arise for existent populations when we do not
know the nature of the parent population but have to estimate some or all
of its characteristics from the data of the sample. In such cases it is
extremely difficult to be completely satisfied that the sampling is random.
Frequently the best we can do is to use a method which has been found
satisfactory for other populations and hope, in the absence of any indica-
tion to the contrary, that it will also be satisfactory for the present
population.

Purposive sampling

16.37 We have already pointed out the dangers of introducing bias
if the observer gives rein to his inclinations in choosing a sample, and
have stressed the fact that in general there does not exist a method of
assessing the degree of accuracy of an estimate made from a purposive
sample. In spite of these handicaps, however, there are cases where
purposive selection is a useful method. In this book we shall not con-
sider it in any great detail, because the reliance placed upon it depends
largely on the circumstances of the case, remains to a great extent a
matter of personal opinion, and is not capable of being discussed by
elementary methods. Nevertheless, our brief survey would be incomplete
without some reference to it.

16.38 Let us first of all consider the case of an observer who wishes
to take a sample of two or three turnips from a cart-load. A random
sample might give us several very large or very small turnips, though it
is unlikely to do so. But if we allow the observer to run his eye over the
whole load and then choose, he is most likely to take what he regards as
average turnips — ^i.e. average in size, weight, shape, and whatever other
quality may be in his mind.

It may be claimed, with some plausibility, that this purposive method
is more likely to give us a sample which is typical or representative of the
population than a random method. The random sample may vary widely
from the average, whereas the purposive sample does not. This gives
the latter an advantage as a rule ; but it may be pointed out —

{a) That as the sample becomes larger the random sample becomes
more and more representative of the parent, whereas, owing to bias, the
purposive sample in general does not,

(6) That in many cases the object of the sample is to give us information
about the whole of the population ; the purposive sample might tell us
more about the mean weight of the turnips, but would probably give a

l^RELIMINARY NOTIONS ON SAMPLING 383

worse idea of the variance of the weights because the observer has
deliberately chosen values near the mean.

16.39 If we had to choose between pure random sampling and pui posive
sampling, our choice would probably be determined by balancing the
uncertainties of the former, which are mainly due to fluctuations of
chance, and the uncertainties of the latter, which are mainly due to bias.
In practice, however, it is often possible to combine the two methods
in stratified sampling and gain some of the advantages of each while
minimising their disadvantages.

The essentials of this process he in dividing the parent population into
strata and taking a random sample from each stratum. For instance, if
we are taking a sample of earned incomes, we might first group individuals
into classes “ earning up to £500 per annum,” earning from £500 to
£1,000 per annum,” and so on, and then choose a random sample from each
class. Or, if we wanted a sample of farms in Great Britain, we might first
classify them roughly as ” devoted mainly to arable crops,” '' devoted
mainly to milk production,” ” devoted mainly to vegetable growing,” etc.,
and again take a random sample from each group.

16.40 Finally, we may also sample a population by first of all arranging
its individuals in groups. This amounts to taking a different sampling
unit. For instance, in sampling the population of Great Britain we might,
as a matter of convenience, take streets or local government districts
instead of individual human beings as our unit. We have already had an
instance of this type when we suggested as one way of sampling a sack of
flour that it might be weighed out first into one-ounce packets. The
process is obviously more convenient when this grouping has been done
for us, e.g., in census returns.

16.41 Each branch of science and industry presents its own sampling
problems, and it would be difficult to expand the foregoing discussion so as
to include the detailed requirements of the worker in every sphere. We
shall revert to the general subject of sampling in Chapter 23, and conclude
this chapter with an example of the way in which all the methods we
have described may be pressed into service in order to give a sample
which is as representative as practical limitations will allow.

It is the practice in England for manufacturers of sugar from sugar beet
to pay the growers according to the sugar* content of their product. The
beet, which is not unlike a parsnip, is delivered to the factory in lots of at
least several tons with a certain amount of waste material, such as earth,
adhering to it. The problem is, then, {a) to find the net weight of the beet
when cleaned and ready for the slicing process, which is the first stage in
the extraction of the sugar, and {b) to ascertain the sugar content. The
method of procedure is as follows —

The gross weight of the load of beet usually is first obtained by weighing
the lorry which contains it when full, and when empty. From the middle

384

THEORY OF STATISTICS

of the load of beet is then abstracted about 28 pounds, which is carefully
weighed, and then cleaned and weighed again. The difference m the
weights gives the tare/' that is to say, the proportion of waste matter,
and a proportional amount is deducted from the whole load to give the
net weight of beet. This process is equivalent to taking a random sample
and assuming that the value of the '' tare " in the sample is the value in
the whole population.

The sample of washed beet is then laid out on a table and arranged with
the roots in order of size. From this sample a smaller sample is taken by
choosing a-beet every so often. This is a process of pure purposive selection.

The reduced sample is still inconveniently large, so it is reduced by
taking a slice from each beet. It is known that the sugar in the root is not
distnbuted homogeneously (although it is roughly symmetrical about the
axis of the root), so trained men are employed to slice one section with a
rasp, the section being that which would be obtained by cutting the root
from the thick end to the tapered end into two symmetrical halves and then
repeating the process one or more times. This selection again is pur-
posive in so far as the shape of the section is based on knowledge of the
distribution of the sugar, but random in so far as it is a matter of chance
what is the longitude of the particular slice chosen.

When each beet has been treated in this way there is given a heap of
pulp which may be analysed. The heap is, however, as a rule still too
large. It is therefore well mixed and divided into four heaps. Two heaps
are thrown away, one is reduced to 26 grammes and analysed by the factory
and one, similarly reduced, is analysed by the grower's representative.
This last method of selection is a random method adapted for a population
which cannot readily be enumerated.

The final sample therefore appears as the .result of foui successive
sampling methods, two of which are random, one purposive, and one a
mixture of purposive and random.

SUMMARY

1. Sampling may be random, purposive or mixed.

2. Random sampling owes its importance to the fact that we can assess
the results obtained from it in terms of probability.

3. The presence of an element of choice on the part of the observer
introduces the danger of bias, and should not be permitted where it can be
avoided.

4. Random samples may conveniently be drawn by the use of card
populations or of random sampling numbers.

5. The sampling technique adopted in any given case will depend largely
on the circumstances of that case and the resources of the observer. At
the present time the reliability of estimates made from samples is partly a
matter of individual opinion founded on intuitive ideas, unless the sampling
methods are random.

PRELIMINARY NOTIONS ON SAMPLING

385

EXERCISES

16.1 Draw a random sample of 20 from the population of men of the last
column of Exercise 4.6 (inhabitants of the United Kingdom classified
according to weight). Find the mean of the sample and compare it with
the mean of the population.

16.2 Deal yourself a hand of 13 cards from an ordinary pack of 52 playing
cards and count the number of court cards. Use your result to estimate
the number of court cards in the whole pack.

Repeat the experiment ten times, taking a new deal each time, and com-
pare the mean of your results with the true value, 12.

16.3 Suggest a method for obtaining a random sample of words from the
English language by the use of random sampling numbers and a dictionary.

16.4 Draw a sample of 30 from the population of the last column of
Table 4.7, and find the standard deviation. Compare your result with the
standard deviation of the population.

16.5 Suggest a possible source of bias in the following —

{a) A barrel of apples is sampled by taking a handful from the
top.

(b) A mixture of sand and sawdust is sampled by scooping up
a quantity from the bottom.

(c) A set of digits is taken by opening a Telephone Directory at
random and choosing the telephone numbers in the order in
which they appear on the page.

(d) Readers of a newspaper are sampled by printing m it an
invitation to them to send up their observations on some
topical event.

{e) Investigators into the size of families in a town conduct a
house-to-house inquiry (1) in the morning, (2) in the after-
noon, ignoring those houses at which there is no reply.

16.6 Draw 100 samples of 10 from a normal population by means of
random sampling numbers, and form the frequency-distribution of their
means.

16.7 In the data obtained in Exercise 16.6, form the frequency-distribu-
tion of the root-mean-square deviations of the samples about the mean
of the parent population.

16.8 Draw 100 samples of 10 from the Poisson population of 8.47, page 194,
and form the frequency-distribution of their means.

16.9 Draw 500 samples of 4 from the population of Australian marriages
of Table 4.8, page 84, and form the frequency-distribution of their range.
16.10' Draw a sample of 50 from the population of Table 9.4, page 204
(4912 dairy cows), and find the correlation m the sample between age in
yeai’S and yield of milk per week. Compare your result with the correla-
tion in the population.

CHAPTER SEVENTEEN

THE SAMPLING OF ATTRIBUTES

LARGE SAMPLES

The problem

17.1 In dealing with the theory of sampling we shall find it convenient
to preserve the formal distinction between attributes and variables
which we drew earlier in this book. The theory of the sampling of
attributes is in many respects simpler than that of variables, and in this
chapter we shall confine ourselves to it. We shall begin by considering
a type of sampling which we shall call simple, involving certain limitations
on the generality of the problem, and shall then proceed to examine the
removal of these limitations in order to deal with the general case.

17.2 The sampling of attributes may be regarded as the drawing of
samples from a population containing A's and not--4's. The number of
^'s in each sample, or the proportion of i4's, will form part of the data
provided by the samples.

We shall find it convenient to adopt the nomenclature of 8.3 and to
speak of the drawing of an individual on sampling as an event.” The
appearance of the attribute A may be called a ” success ” and the non-
appearance a ” failure.” Thus, in sampling a human population for the
proportions of the two sexes, we might say of a sample of 100, 45 of which
were male, that the sample consisted of 100 events, 45 of which were
successes and 55 failures. (It might, of course, be more convenient —
and would certainly be more courteous — ^to reverse the names and call
the occurrence of a female a ” success ” and of a male a ” failure.”)

Simple sampling

17.3 By simple sampling we mean random sampling in which each
event has the same chance p of success, and in which the chances of
success of different events are independent, whether previous trials have
been made or not. These conditions hold good, for instance, in the
throwing of a die or the tossing of a coin ; the chance of getting heads
with a coin is not affected by what was obtained on the previous trials,
and remains constant no matter how many trials are made, provided, of
course, that the coin does not begin to wear or is not falsely manipulated
by the experimenter.

Simple sampling is a particular form of random sampling, as we have

386

THE SAMPLING OF ATTRIBUTES

387

defined it in the previous chapter. Suppose, for example, we take a
sample of two from a population consisting of 6 men and 4 women under
random sampling conditions, i.e. so that at each of the two events which
constitute the sample every member of the population has an equal chance
of being chosen. If, at the first trial, we draw^ a man, the chance of doing
so being ®o> there will be 5 men and 4 v/omen left in the population, and
the chance of obtaining a man on the second trial will be | . This is not
the same as the chance on the first trial, and hence the sampling is not
simple, though it is random.

Mean and standard deviation in simple sampling of attributes

17.4 Suppose now that we take N samples with n events m each. The
chance of success of each event is p and of its failure q—\ —p. As in
8.6, the frequencies of samples with 0, 1, 2, . . . successes are the terms
in the series i.e.

. . . J^nqp*^-'^ -^P^

As in 8.9, this distribution has mean M given by

M=np

and standard deviation (8.10)

(jz=^y/npq .... (17,1)

17.5 In lieu of recording the number of successes in each sample we

might have recorded the proportion of successes, that is, -th of the

number in each sample. As this would amount to dividing ail figures
of the record by n, the mean proportion of successes must be p, and the
standard deviation of the proportion of successes is given by

“"Vf (>“)

Equations (17.1) and (17.2) are of fundamental importance.

Example 17.1. — The following results, due to Weldon, are of interest.
Weldon threw 12 dice 4,096 times, a throw of 4, 5 or 6 being called a
success. We have, then, 4,096 samples of 12 from the population con-
sisting of all possible throws of the dice.

If the dice are all true, the chance of success is Hence, the theoretical
mean M =6 ; theoretical value of the standard deviation a = y^O '5x0*5x12
=1*732.

388

THEORY OF STATISTICS

The following was the frequency-distribution observe'd —

Successes

Frequency

Successes

Frequency

—

847

536

257

198

430

731

—

948

Total

4,096

Mean ikf =6 *139, standard deviation 0 = 1*712. The proportion of
successes is 6*139/12—0*512 instead of 0*5.

Example 17.2, — (G. U. Yule.) The following may be taken as an illustra-
tion based on a smaller number of observations : Three dice were thrown
648 times, and the numbers of 5's or 6’s noted at each throw. ^—1/3,
2/3 ; theoretical mean 1 , standard deviation 0*816.

Frequency-distribution observed —

Successes Frequency

0 179

1 298

2 141

3 30

Total 648

M== 1*034, OS'— 0*823. x^ctual proportion of successes 0*345.

17.6 The value pn is sometimes called the expected value of the
number of successes in the sample. It is not only the mean value of
ail samples, but is the most probable value and is also representative, i.e.
it bears the same ratio p to the number in the sample as the number of
individuals with attribute A in the population bears to the total number
in the population. The divergences of the number of successes from the
expected value in any given random sample give rise to what we have
hitherto called fluctuations of random sampling. They are to be regarded
as deviations due to the nature of the sampling process, and not indicative
of any real properties of the population itself.

17.7 Equations (17.1) and (17.2) enable us to deal with the question
which has arisen several times in earlier chapters of this book, namely,
when can we say that observed deviations from the expected values in
a sample of attributes are due to some real effect and are not merely
attributable to sampling Puctuations ?

The binomial distribution, to v/hich samples classified according to
the frequencies of an attribute give rise, is a single-humped type which
approximates very closely to the normal for large values of n, the number

THE SAMPLING OF ATTRIBUTES

389

in the sample. It follows that the great majority of its members lie
within a range ±30- on each side of the mean, i.e. of ±^'\/npq on each
side of the value np. If the distribution is exactly normal, 0*9973 of the
curve lies within this range (8.29). We can therefore say that if a
particular sample gives a value of p outside this range, the deviation from
the expected value is most unlikely to have arisen from fluctuations of
simple sampling. If n is large, the chances are about 3 in a thousand
that it arose in that way.

It must be emphasised that the free use of the Sa rule is justified only
if n is large.

Example 17.3. — In the experiments of Example 17.1, 25,145 throws of
a 4, 5 or 6 were made out of 49,152 throws altogether. The chance of
throwing one of these numbers is and hence the expected value is 24,576.
The observed number was thus 569 in excess of this. Can the deviation
from the expected value be due to fluctuations of simple sampling ?

The stancjard deviation of simple sampling is

a=Vnpq = V^xix49\52
=110*9

The deviation observed is 5*13 times this quantity, and it is therefore
most improbable that it arose as a sampling fluctuation. We must there-
fore seek some other explanation of the deviation, and it seems reasonable
to suspect that the dice were slightly biased.

The problem might, of course, have been attacked equally well from
the standpoint of proportion instead of the actual numbers of successes.
This proportion is 0*5116 instead of the expected 0*5000, the difference
in excess being 0*0116. The standard deviation of the proportion is

and the difference observed is 5*13 times this, which is the same ratio as
before, as of course it must be.

Example 17.4. — (Data from the Second Report of the Evolution Com--
mittee of the Royal Society, 1905, p. 72.)

Certain crosses of the pea, Ptsum sativum, gave 5,321 yellow and 1,804
green seeds. The expectation is 25 per cent of green seeds on a Mendelian
hypothesis. Can the divergences from the expected values have arisen
from fluctuations of simple sampling only ?

The numerical difference from the expected result is 23. The standard
deviation of simple sampling is

a= Vo *25 X 0 *75 X 7125 =36*6

The divergence from theory is only about 0 • 6 of this, and hence may
very well have arisen from fluctuations of simple sampling.

390

THEORY OF STATISTICS

Standard error

17.8 We shall very frequently have to use the standard deviation of
sampling, and it is convenient to have a shorter name for this quantity.
We shall call it the standard error. The use of the word error is justified
in this connection by the fact that we usually regard the expected value
as the true value, and divergences from it as errors of estimation due to
sampling effects ; but the student should not attach too much significance
to the particular term “ error."'

In most of our work the term standard error " will be applied to the
standard deviation of simple sampling ; but it has a rather wider meaning,
embracing this one, which we shall discuss in considering the sampling of
variables (18.22, cf. also 17.31).

We may, then, summarise the foregoing in the statement that fre-
quencies differing from the expected frequency by more than 3 times the
standard error are almost certainly not due to fluctuations of sampling.
They point to some departure of the sampling from simplicity, which may
in turn point either to some flaw in the sampling technique or to causal
effects in the population itself.

Probable error

17.9 Instead of the standard error, some authorities have used a quantity
called the probable error, which is 0 • 67449 times the standard error. This
practice arose from the fact that in the normal curve the quartiles are
distant 0*67449o- from the mean, so that the probability that a deviation
is in excess of the probable error is and is equal to the probability of a
deviation being less than the probable error. The rule that the observed
deviation should not be greater than 3 times the standard error is then
approximately equivalent to a rule that it should not exceed 4-5 times
the probable error.

The use of the probable error is declining, and we recommend the student
to eschew it.

17.10 In Examples 17.1 to 17.4 we dealt with cases where p, the
probability of success, was known a priori. In many cases it is not known,
and further consideration is necessary before we can apply equations (17.1)
and (17.2) to such cases.

To fix the ideas, let us suppose that we have a simple sample of 1,000
individuals from the inhabitants of Great Britain, and find that 36 per cent
of them have blue eyes and the remainder have eyes of some other colour.
What can we infer about the proportion of blue-eyed individuals in the
whole population ?

In this instance we do not know the proportion p of blue-eyed in-
dividuals in the population. We do know that the standard error is
VlOOO^y Now, whatever and q are, pq cannot exceed and hence the
standard error cannot exceed Ia/IOOO, or 16. Hence, whatever p is, a

THE SAMPLING OF ATTRIBUTES

39 ^

simple sample should give a number of successes within 3 times this, or 48,
of the expected frequency pn. This is 4 - 8 per cent of the sample, and we
thus may say that the proportion of blue-eyed people in the whole popula-
tion is 36±4‘8 per cent, i.e. that it lies between 31*2 and 40*8 per cent

17.11 We may, however, make a rather better estimate. We have
seen that the standard error is small compared with the expected value,
and hence with the observed value. If, therefore, in calculating the
standard error we take the observed values of p and q in the sample instead
of the unknown true values of p and q, we shall not involve ourselves in
very great error.

Thus, taking p to be 0*36, g==0*64,

CT == V = Vo -36 X V64 xlOOO
=15-18

Hence, 3a=45*5 approximately, and the limits are now 36^4*6 or
31*4 and 40*6 — slightly narrower than those previously obtained.

17.12 In this example we have taken the proportion of successes in
the sample to be an estimate of the proportion of successes in the popula-
tion, and have set limits to the range within which the true proportion
probably lies. There are other reasons, of an advanced theoretical
character which we shall not specify, for taking p in the sample as an
estimate of p in the population, but the student will probably concede
that it is the most reasonable thing to do in the circumstances. We must,
however, look a little more closely into the assumption that this estimate
may be used in calculating the standard error.

17.13 The assumption is a justifiable one if n is large and neither p nor
q is small. For in such a case, the standard error of the proportion p is

4 '

'Pq

and this is small compared with p unless p itself is small.

If, then, the standard error of p is small, the value of p estimated from
the sample must be close to the real value, and we shall not introduce any
sermus error by taking the estimated value in evaluating the formula

fpi.

y ^

17.14 Precisely how large n must be for this approximation to be valid
it is not easy to say. Samples of 1,000 are almost pertainly large enough,
and we may often apply the foregoing procedure with considerable
confidence to much smaller samples, say of 100. For samples below that
figure it IS as well to examine carefully the circumstances of any given case
and to proceed with caution.

We shall have more to say on this matter when we consider the sampling
of variables (18.17 and 18.18).

39 ^

THEORY OF STATISTICS

For the remainder of this chapter we shall assume that our samples
are “ large/' that is to say, that the approximations involved in our
assumptions as to the estimate of p are valid.

Example 17.5. — A sample of 900 days is taken from meteorological
records of a certain district, and 100 oi them are found to be foggy. What
are the probable limits to the percentage of foggy days in the district ?

Anticipating somewhat our discussion of simple sampling, we will
assume that the conditions of this problem give a simple sample.

Hence,

p=^\>

Standard error of the proportion of foggy days

_ fM_ /I 8
V« y 9^9^900

=0-0105

=1*05 per cent.

Hence, taking ] to be the estimate of the number of foggy days, we have
that the limits are 11*11 per cent ±3*15 per cent, i.e. 8 per cent and
14*25 per cent approximately.

Example 17.6. — A biased penny is tossed 100 times and comes down
heads 70 times. What are the probable limits to the probability of getting
a head in a single trial ?

We require to know the limits of p. If we assume that 100 is a large
sample, we have —

The limits are therefore 0*70±(3x 0*0458)

=0*70±0*1374

=0*56 and 0*84 approximately

If we feel any doubt as to the validity of using estimates of p and q
from a sample of 100 in calculating the standard error, we may proceed
as follows —

The standard error of p cannot exceed i.e. 0*05. Hence

the value of p lies almost certainly within the limits 0*70 ± 0*15, i.e. 0*55
and 0*85.

11^=0*55,

0=0-04975

^^= 0-03571

If ^=0-85,

THE SAMPLING OF ATTRIBUTES

393

For intermediate values oi p tl lies between these limits. Hence the

V ^

maximum value of the standard error is 0*04975, and p lies between the
limits 0*70 d: 0*14925, i.e.

0*55075 and 0*84925

It will be seen that these limits are nearly equal to those obtained on
the assumption that and are not very different from those we

got by assuming _^=0-70. There would, however, be an appreciable
difference if p had been small, say 0*10.

17.15 If one of the two proportions p and q becomes very small, equation
(17.1) may be put into an approximate form that is very useful. Suppose
p to be the proportion that becomes very small, so that we may neglect
compared with p ; then

pq:=:^p ^p^ z=:p approximately
and consequently we have approximately —

G=^vUtp = ^/M • . . . (17.3)

That is to say, if the proportion of successes he small, the standard
deviation of the number of successes is the square root of the mean number
of successes. Hence we can find the standard error even though p be
unknown provided only we know that it is small.

This is, in fact, the case when the binomial becomes the Poisson series
(8.40). For such distributions the rule that a range of 6a includes the
great majority of the observations remains valid, as may be seen from
the diagram on page 192, but the limits assigned to the standard error of
the mean M may be too wide on the left of the mean. For example, if
M=l, a— 1, and a range of 3 units to the left of the mean carries us to a
value of —2, whereas there can be no part of the frequency with negative
values of the variate.

17.16 It will be noticed that the standard error depends only on the
value of p and the size of the sample, and that therefore the range within
which p probably lies is independent of the size of the population. This
appears a little paradoxical, because one might expect that a sample
which was, say, 20 per cent of the population would enable closer limits
to be set than one which was 10 per cent of the population. The ordinary
man nearly always believes that a sample of only 1 /1 000 of the population
necessarily give§ much less trustworthy results than a sample of say, 1 /lO,
without regard to its actual size, but the belief is quite unjustified.

The explanation is to be found in the nature of simple sampling itself.
We shall see overleaf that the conditions under which simple sampling arises
in practice are such that either the population is actually or practically
infinite, or each member drawn for a sample is put back in the population

394

THEORY OF STATISTICS

before the next is drawn. In either case the population is inexhaustible,
and no sample is any nearer to including ail its members than another
sample. It is, therefore, not surpnsing to find that the size of the popula-
tion does not appear in the formula for the standard error.

17.17 A further notable fact is that the standard error of p varies
inversely as the square root of n, and not inversely as n itself. Thus, as
n becomes larger the standard error becomes smaller, which is what we
should expect, but the standard error decreases proportionately to the
square root of n. For instance, if a sample of 100 gives us a standard
error of 10 per cent, it will take a sample of 400 to halve that error, and
a sample 100 times as large, i.e. 10,000, to reduce the error to one-tenth
or one per cent.

Precision

17.18 The standard error may fairly be taken to measure the unreliability
of an estimate of p ; the greater the standard error, the greater the
fluctuations of the observed proportion, although the true proportion
is the same throughout. The reciprocal of the standard error (1 /s), on
the other hand — or some convenient multiple of the reciprocal — may be
regarded as a measure of reliability, or, as it is sometimes termed, precision,
and consequently the reliabtlity or precision of an observed proportion
varies as the square root of the number of observations on which %t is based.

The limitations of simple sampling

17.19 In order to realise the limitations on the use of the formulae of
equations (17.1) and (17.2), it is necessary to consider what are the con-
ditions which will give rise to simple sampling in practice. Supposing, for
example, that we observe among groups of 1 ,000 persons, at different times
or in different localities, the various percentages of individuals possessing
certain characteristics — dark hair, or blindness, or insanity, and so forth.
Under what conditions should we expect the observed percentages to
obey the law of sampling that we have found, and show a standard
deviation given by equation (17.2) ?

17.20 In the first place, the condition that p, the probability of drawing
an individual with attribute A on random sampling, remains constant,
and in particular is the same for all samples, means that the proportion
of individuals with attribute A in the population must remain constant
at the drawing of each sample. Consequently, if formula (17.2) is to
hold good in our practical case of sampling there must not be a difference
in any essential respect — i.e. in any character that can affect the proportion
observed — between the localities from which the samples are drawn, nor,
if the samples have been made at different epochs, must any essential
change have taken place during the period over which the observations
are spread. Where the causation of the character observed is more or
less unknown, it may, of course, be difi&cult or impossible to say what

THE SAMPLING OF ATTRIBUTES

395

differences or changes are to be regarded as essential, but where we have
more knowledge the condition laid down enables us to exclude certain
cases at once from the possible applications of formula (17.1) or (17.2).
Thus it is obvious that the theory of simple sampling cannot apply to the
variations of the death-rate in localities with populations of different
age and sex composition, or to death-rates in a mixture of healthy and
unhealthy districts, or to death-rates in successive years during a period
of continuously improving sanitation. In all such cases variations due
to definite causes are superposed on the fluctuations of sampling.

17.21 Secondly, the proportion of individuals with attribute A must
remain constant for the drawing of each individual member of the sample.
This is again a very marked limitation. To revert to the case of death-
rates, formulae (17.1) and (17.2) would not apply to the numbers of persons
dying in a series of samples of 1 ,000 persons, even if these samples were all
of the same age and sex composition, and living under the same sanitary
conditions, unless, further, each sample only contained persons of one sex
and one age. For if each sample included persons of both sexes and
different ages, the condition would be broken, the chance of death during
a given period not being the same for the two sexes, or for the young
and the old. The groups would not be homogeneous in the sense required
by the conditions from which our formulae have been deduced.

X7.22 We pointed out in 17.3 that sampling from a finite population
is not simple owing to the fact that the abstraction of an individual alters
the chance of success at the next trial. In practice there are three
important cases in which the condition for the constancy of p is satisfied :

(а) If the individuals are replaced at each drawing before the next
drawing is made ; for in this case the constitution of the population is the
same at each trial, and hence the chance of success must also be the same.

(б) If the population is infinite ; for in this case the withdrawal of a
finite number of members does not affect the proportion of individuals in
the population possessing the attribute in question.

(c) If the population is very large, p may be taken to be constant with-
out sensible error, provided that the sample is not also large. This is a
very important case, and justifies the application of the theory of simple
sampling to many practical data.

Suppose, for instance, we are sampling the population of the United
Kingdom for sex ratio, and decide to take a sample of 1,000. Suppose
again, for the purposes of illustration, that the whole population consists
of 23 million women and 22 million men. The chance of getting a man at

^ ^ ^ ' 1 -11 22 , 000,000
the first trial will then be

If we succeed in getting a man,

45,000,000

21 999 999

the chance of doing so at the second trial will be Even if

draw 999 men the chance of success at the thousandth trial would be

THEORY OF STATISTICS

396

21.999.001

44.999.001 ■

All these chances, to a close approximation, are equal, and we

can assume them to be so without fear of appreciable error. The case
would, of course, have stood differently if our sample had numbered several

millions.

17.23 A third condition for simple sampling was explicitly stated in
our definition in 17,3. The individual events must be completely in-
dependent of one another, like the throws of a die, or sensibly so, like the
drawing of balls from a bag containing a number of bails which is large
compared with the number drawn. Reverting to the illustration of a
death-rate, our formulae would not apply even if the sample populations
were composed of persons of one age and one sex, if we were dealing, for
example, with deaths from an infectious or contagious disease. For if one
person in a certain sample has contracted the disease in question, he has
increased the possibility of others doing so, and hence of dying from the
disease. The same thing holds good for certain classes of deaths from
accident, e.g. railway accidents due to derailment, and explosions in mines :
if such an accident is fatal to one person it is probably fatal to others also,
and consequently the annual returns show large and more or less erratic
variations.

17.24 It is evident that these conditions very much limit the field of
practical cases of an economic or sociological character to which formulae
(17.1) and (17.2) can apply without considerable modification. The
formulae appear, however, to hold to a high degree of approximation in
certain biological cases, notably in the proportions of offspring of different
types obtained on crossing hybrids, and, with some limitations, to the
proportions of the two sexes at birth. It is possible, accordingly, that in
these cases all the necessary conditions are fulfilled, but this is not a
necessary inference from the mere applicability of the formulae. In the
case of the sex ratio at birth it seems doubtful whether the rule applies to
the frequency of the sexes in individual families of given numbers, but it
does apply fairly closely to the sex ratios of births in different localities,
and still more closely to the ratios in one locality during successive periods.
That is to say, if we note the number of males in a series of groups of
n births each, the standard deviation of that number is approximately
Vnpq, where p is the chance of a male birth ; or, otherwise, Vpq In is the
standard deviation of the proportion of male births.

Applications of simple sampling

17.25 We have already shown m examples how the theory of simple
sampling can be used to gauge the precision of an estimate of the proportion
of individuals in a population which possess an attribute A , and to set limits
outside which that proportion probably does not lie. We now turn to
further applications of the theory in the checking and control of the
interpretation of statistical results.

THE SAMPLING OF ATTRIBUTES

397

17.26 Case 1. — Given the expected frequency in a sample and the
observed frequency of successes, it is desired to know whether the deviation
of the second from the first can have arisen from tluctuations of simple
sampling.

This is a case which we have discussed in Examples 17.3 and 17.4.
From the expected frequency we can calculate the standard error, and if
the deviation is more than 3 times this quantity it almost certainly did not
arise from fluctuations of random sampling.

17.27 One caution is necessary here. If the deviation is less than
3 times the standard error, it does not follow that the expected frequency
divided by the number in the sample is really the proportion of individuals
possessing the attribute A in the population. In other words, if the
expected value is derived from some hypothesis, such as the Mendelian
hypothesis in the case of Example 17,4, the fact that the deviation lies
within the limits of 3 times the standard eixor does not prove the hypothesis
correct. It only indicates that experiment and hypothesis are not in
disagreement. Furthermore, if the deviation lay without those limits,
the hypothesis would not necessarily be disproved, for the fault might
lie with the randomness of the sampling.

17.28 Case 2. — Two samples from distinct materials or different popula-
tions give proportions of /Ts and the numbers of observations in
the samples being and respectively, (a) Can the difference between
the two proportions have arisen merely as a fluctuation of simple sampling,
the two populations being really similar as regards the proportion of A's
therein ? (5) If the difference indicated were a real one, might it vanish,
owing to fluctuations of sampling, in other samples taken in precisely the
same way ? This case corresponds to the testing of an association which is
indicated by a comparison of the proportion of amongst B's and fi's.

(a) We have no tlieoretical expectation in this case as to the proportion
of in the population from v/nich either sample has been taken.

Let us And, however, whether the observed difference between and
p 2 may not have arisen solely as a fluctuation of simple sampling, the
proportion of A 's being really the same in both cases, and given, let us say,
by the (weighted) mean proportion in our two samples together, i.e. by

pQ —

{the best guide that we have).

Let fig be the standard errors in the two samples, then

If the samples are simple samples in the sense of the previous work, then
the mean difference between and p^ will be zero, and the standard error

398

THEORY OF STATISTICS

of the difference the samples being independent, will be given by

.... (n.4)

If the observed difference is less than some tliree times 63 ^ 2 ^ have

arisen as a fluctuation of simple sampling only.

(Z>) If, on the other hand, the propoitions of A's are not the same in the
material from which the two samples are drawn, but pj, and are the true
values of the proportions, the standard errors of sampling in the two cases
are

= Piii K. jn^

and consequently

.... (17.5)

If the difference between p^ and p^ does not exceed some three times
this value of 632 , it may be obliterated by an error of simple sampling on
taking fresh samples in the same way from the same material.

The student will note that in arriving at these results we have assumed
that the unknown values pQ, pi, p 2 , are given to a sufficient degree of
approximation by estimates from the samples. This, as we have seen, is
justified if n be large.

Example 17.7. — (Data from J. Gray, '' Memoir on the Pigmentation
Survey of Scotland,'* Jour, of the Royal Anthropological Institute, 1907,
37). The following are extracted from the tables relating to hair-colour
of girls at Edinburgh and Glasgow —

Edinburgh

Glasgow

Of medium Total Per cent

hair-colour observed medium

4,008 9,743 4M

17,529 39,764 44*1

Can the difference observed in the percentage of girls of medium hair-
colour have arisen solely through fluctuations of sampling ?

In the two towms together the percentage of girls with meditim hair-
colour is 43*5 per cent. If this were the true percentage, the standard
error of sampling for the difference between percentages observed in
samples of the above sizes would be —

= 0*56 per cent.

The actual difference is 3 • 0 per cent, or over 5 times this, and could not
have arisen through the chances of simple sampling.

THE SAMPLING OF ATTRIBUTES

399

If we assume that the difference is a real one and calculate the standard
error by equation (17.5), we arrive at the same value, viz, 0-56 per cent.
With such large samples the difference could not, accordingly, be
obliterated by the fluctuations of simple sampling alone.

17.29 Case 3. — Two samples are drawn from distinct material or different
populations, as in the last case, giving proportions of A's and p 2 , but
in lieu of comparing the proportion with p^ it is compared with the
proportion of ^'s in the two samples together, viz. p^, where, as before.

Required to find whether the difference between p^ and p^ can have arisen
as a fluctuation of simple sampling, p^ being the true proportion of ^’s
in both samples.

This case corresponds to the testing of an association which is indicated
by a comparison of the proportion of A*s amongst the JB's with the pro-
portion of A 's in the population. The general treatment is similar to that
of Case 2, but the work is complicated owing to the fact that errors in
p^ and pQ are not independent.

If be the standard error of the difference between p-j^ and p^, we
have at once —

Cqi =

being the correlation between errors of simple sampling in pi and p^.
But from the above equation relating pQ to p^ and p^, writing it in terms
of deviations in p-^ and p^, multiplying by the deviation in p^ and
summing, we have, since errors in p^ and p^ are uncorrelated —

%+% €o V %+^2

Therefore finally —

^2 ^ ... (17.6)

Unless the difference between p^ and p^ exceed, say, some three times
this value of Coi, it may have arisen solely by the chances of simple
sampling.

It will be observed that if n-^ be very small compared with %
approaches, as it should, the standard error for a sample of observations.

We omit, in this case, the allied problem whether, if the difference
between p^ and p^ indicated by the samples were real, it might be wiped
out in other samples of the same size by fluctuations of simple sampling

400

THEORY OF STATISTICS

alone. The solution is a little complex, as we no longer have

Example 17.8. — Taking now the figures of Example 17.7, suppose
that we had compared the proportion of girls of medium hair-colour in
Edinburgh vvdth the proportion in Glasgow and Edinburgh together.
The former is 41 * 1 per cent, the latter 43 • 5 per cent, difference 2 • 4 per cent.
The standard error of the difference between the percentages observed in
the sub-sample of 9,743 observations and the entire sample of 49,507
observations is, therefore,

/ 39,764

e„,=(43-5x56.5)if^jg^^^j =.0-45 per cent.

The actual difference is over five times this (the ratio must, of course, be
the same as in Example 17.7), and could not have occurred as a mere
error of sampling.

Effect of removing the limitations of simple sampling

17,30 Let us now consider the effect on the standard error of the removal
of the conditions of simple sampling which we discussed in 17.19 to 17.24.

The breakdown of the condition we discussed in 17.20, namely, that
the proportion of A's in the population should remain constant for aU
samples, might occur if we took a number of samples from a changing
population or from different strata of a population which was not homo-
geneous.

We may represent such ciicumstances in a case of artificial chance by
supposing that for the first throv/s of ii dice the chance of success for
each die is p^, for the next /g throws p^, for the next/g throws p^, and so
on, the chance of success varying from time to time, just as the chance
of death, even for individuals of the same age and sex, varies from district
to district. Suppose, now, that the records of all these throws are pooled
together. The mean number of successes per throw of the n dice is given

^ . . . ) — npQ

where N^'L{f) is the whole number of throws, and is the mean value
S(/j^) jN of the varying chance^. To find the standard deviation of the
number of successes at each throw, consider that the first set of throws
contributes to the sum of the squares of deviations an amount

^Pi9i being the square of the standard deviation for these throw^s, and
^ipi^po) the difference between the mean number of successes for the
flist set and the mean for all the sets together. Hence the standard

THE SAMPLING OF ATTRIBUTES

401

deviation a of the whole distribution is given by the sum of all quantities
like the above, or

Let o-j, be the standard deviation of p, then the last sum is and

substituting 1 —p for q, we have —

==np^qQ+n{n-\)(j^^ .... (17.7)

This is the formula corresponding to equation (17.1) ; if we deal with
the standard deviation of the proporiton of successes, instead of that of
the absolute number, we have, dividing through by n^, the formula
corresponding to equation (17.2), viz. —

A?o , n-l

s2 -f.

(17.8)

17.31 If n be large and Sq be the standard error calculated from the
mean proportion of successes pQ, equation (17.8) is sensibly of the form

We have thus analysed into two parts, Sq^ the portion due to devia-
tions from the mean p^, and the portion due to variations of the p*s
about their mean. The former we may regard as the contribution to
5 ® due to chance fluctuations ; the latter as the contribution due to real
variation of the proportions among the different strata of the population.

In conformity with later work we shall continue to call 5 (or a if we
are dealing with frequencies) the standard error, although the sampling
is no longer simple. The deviation s is still, in fact, the standard deviation
of the various sample values of p about the mean value. The term

(or VnpQqQ), on the other hand, is what the standard error would have
been if the sampling had been simple, and from the above equation we
accordingly see that the eflect of the breakdown of the first condition for
simple sampling is to increase the standard error.

We may illustrate the effect of variations in p on the data of Table 17.1,
showing the percentages of the electorate voting in municipal elections
in England, in various groups according to size of electorate. (The
figures in the original returns for percentages are given to the first place
of decimals, so the intervals are centred at 20*45, 27*45, etc.)

At the foot of the table we show the actual variances and the
theoretical variances based on the formula pq In. For instance, in the
size group 0 — 5,000 we have p =0*5621 and take n as the mid-point
of the range, namely 2,500. The variance (in terms of percentages, not
proportions) is then (0*5621 X 0*4379 x 100^) /2, 500 =0*98.

Now it is clear from these data that the theoretical variances are only
a very small proportion of the actual variances. In short we cannot

402

THEORY OF STATISTICS

assume, even in electorates of about the same size, that the numbers
voting are distributed in the binomial form. There is, so to speak, no
proneness to vote’' common to all electors and represented by the
proportion p. There are (as we know for elections) substantial variations
between electorates, represented by the variances

The effect of these results on straw votes ” for the forecasting of
elections is evident. We cannot measure the standard error of proportions
in samples of persons indicating their voting intentions by the simple-
sampling formulae.

TABLE 17.1. — Percentages of electorate voting in municipal elections in England in 1945

County boroughs and boroughs with more than 100,000 voters omitted. Electorate "
includes only those persons entitled to vote on this occasion, i e., persons in non-
contested areas are excluded.

Data from Registrar-General’s Review of England and Wales for 1946, Tables Part II Civil.

Percentage of electorate
voting

5,000

Size of electorate

5,001 10,001 15,001

to to to

10,000 15,000 20,000

20,001

50.000

50,001

100,000

20-

25-

30-

35-

40-

45-

50-

55-

—

60-

—

65-

—

70-

—

75-

—

oU—

85-

90-

—

Totals .

433

272

162

156

Means . . . i

56*21

50*51

48 81

47*83

45*56

39*79

Variances s^.

120*12

113*45

111*43

140*36

82*80

85*91

Theoretical variances

0*98

0*33

0*20

0*14

0*07

0*03

V(s»-V) •

10*9

10*6

10*5

11*9

9*1

9*3

The figures of this case also bring out clearly one important consequence
of (17.8), viz. that if we make % large, s becomes sensibly equal to a,,,
while if we make n small, s becomes more nearly equal to po^oln. Hence,
if we 'want to know the significant standard deviation of the proportion p
— the measure of its fluctuation owing to definite causes — n should be
made as large as possible : if, on the other hand, we want to obtain good
illustrations of the theory of simple sampling, n should be made small.
If n be very large, the actual standard error may evidently become almost
indefinitely large compared with the standard deviation of simple sampling.

THE SAMPLING OF ATTRIBUTES

403

Thus during the twenty years 1855-74 the death-rate in England and Wales
fluctuated round a mean value of 22*2 per thousand with a standard
deviation (s) of 0’86. Taking the mean population as roughly 21 millions,
the standard deviation of simple sampling (sq) is approximately

= 0*032 per thousand
y 21x10^ ^

This is only about one twenty-seventh of the actual value.

17,32 Now consider the effect of altering the second condition of simple
sampling dealt with in 17.21, viz. the circumstances that regulate the
appearance of the character observed shall be the same for every in-
dividual or every sub-class in each of the populations from which samples
are drawn. Suppose that in a group of n dice thrown the chances for
dice are ; for Wa dice, ^.nd so on, the chances varying for

different dice, but being constant throughout the experiment. The case
differs from the last, as in that the chances were the same for every die
at any one throw, but varied from one throw to another ; now they are
constant from throw to throw, but differ from one die to another as they
w'ould in any ordinary set of badly made dice. Required to find the effect
of these differing chances.

For the mean number of successes we evidently have —

= npQ

pQ being the mean chance 'L{mp) jn. To find the standard deviation of the
number of successes at each throw, it should be noted that this may be
regarded as made up of the number of successes in the dice for which the
chances are p^.q^, together with the number of successes amongst the m 2
dice for which the chances are p^.q^, and so on ; and these numbers of
successes are all independent. Hence,

= ^^{mpq)

Substituting 1 —p for q, as before, and using to denote the standard
deviation of p,

cr2 np^q^—w^^ .... (17.9)

or if s be, as before, the standard error of the proportion of successes,

n n

(17.10)

Hence, in this case the standard error s is less than the standard error
of simple sampling.

404

THEORY OF STATISTICS

17.33 The extent to which the standard error is affected may con-

ceivably be considerable. To take a limiting case, if p be zero for half the
events and unity for the remainder, and so that s is zero.

To take another illustration, still somewhat extreme, if the values of p
are uniformly dislnbuted over the whole range between 0 and 1,

as before, but cr 3 >^=l /12=0*0833 (6.15, p. 136). Hence, 0* 1667/^,
j =0* 408/ instead of 0-5/vV/, the value of s if the chances are ^ ineverj''
case. In most practical cases, however, the effect will be much less. Thus
the standard de\dation of simple sampling for a death-rate of, say, 14 per
thousand in a population of uniform age and one sex is (14 x986)^/VtJ
=118/V^ S' population of the age composition of that of England
and Wales, however, the death-rate is not, of course, uniform, but varies
from a high value in infancy (say 64 per thousand), through very low
values (2 to 3 per thousand) in childhood to continuously increasing values
in old age ; the standard deviation of the rate within such a population
is roughly about 24 per thousand. But the effect of this variation on the
standard deviation of simple sampling is quite small, for, as calculated from
equation (17.10),

S* = 1(14 x 986 -576)

s =

as compared with 118/V«.

17.34 We have, finally, to pass to the condition referred to in 17.23,
and to discus^ the effect of a certain amount of dependence between the
several events in each sample. We shall suppose, however, that the
two other conditions are fulfilled, the chances p and q being the same for
every event at every trial, and constant throughout the experiment. The
standard deviation for each event is [pq)^ as before, but the events are no
longer independent ; instead, therefore, of the simple expression

0-2 = npq

we must have (cf. 14.2, p. 327)

a* =«^?+2Mri5i+>'i8+ • . . +ri3+ • • ■)

where etc. are the correlations between the results of the first and

second, first and third events, and so on — correlations for variables (number
of successes) which can only take the values 0 and 1, but may neverthe-
less be treated as ordinary variables. There are w(w— 1)/2 correlation
coefficients, and if, therefore, r is the arithmetic mean of the correlations,
we may write —

a® = npq\\ 1)]

. (17,11)

THE SAMPLING OF ATTRIBUTES

405

The standard deviation of simple campling will therefore be increased or
diminished according as the average correlation between the results of
the single events is positive or negative, and the effect may be considerable,
as a may be reduced to zero or increased to n{pq)^. For the standard
deviation of the proportion of successes in each sample we have the
equation

=^?[l+r(«_l)] .... (17.12)

17.35 It should be noted that, as the means and standard deviations
for our variables are all identical, r is the correlation coeffxient for a table
formed by taking all possible pairs of results in the n events of each sample.

It should also be noted that the case when r is positive covers the
departure from the rules of simple sampling discussed in 17 . 30 - 17 . 31 ;
for if we draw successive samples from different records, this inlroduces
the positive correlation at once, even although the results of the events at
each trial are quite independent of one another. Similarly, the case dis-
cussed in 17.32-17.33 is covered by the case when r is negative ; for if
the chances are not the same for every event at each trial, and the chance
of success for some one event is above the average, the mean chance cf
success for the remainder must be below it. The present case is, however,
best kept distinct from the other two, since a positive or negative correlatiora
may arise for reasons quite different from those discussed in 17 . 30 - 17 . 33 *

17.36 As a simple illustration, consider the important case nf samphng
from a limited population, e.g. of drawing n bills ii: siiccessiun fr-j'U the
whole number w in a bag containing pw white balls and jr* blac^

On repeating such diawmgs a large number of times, w^e are e\ddeiat]y
equally likety to get a white ball or a black ball for the first, second or nth
bail of the sample ; the correlation table formed from all possIMe pairs of
every sample will therefore tend in the long run to give just the same form
of distribution as the correlation table formed from all posbible pairs of
the w balls in the bag. But from 11.41, page 276, we know tiiat the
correlation coefficient for this table is — 1 /(z^—1), whence

If we have the obviously correct result that <J={pq)^i as in drawl-
ing from unlimited material ; if, on the other hand, o‘ becomes zero
as it should, and the formula is thus checked for simple cases. For draw-
ing 2 balls out of 4, a becomes 0-816 {npq)i ; for drawing 5 balls out of
10, 0’745{npq )^ ; in the case of drawing half the balls out of a very large
number, it approximates to iO^Snpq)^, or 0*707{npq)i.

4o6

THEORY OF STATISTICS

17,37 In the case of contagious or infectious diseases, or of certain
forms of accident that are apt, if fatal at all, to result in wholesale deaths,
r is positive, and if n be large (as it usually is in such cases), a very small
value of r may easily lead to a very great increase in the observed standard
deviation. It is difficult to give a really good example from actual
statistics, as the conditions are hardly ever constant from one year to
another, but the following will serve to illustrate the point. During the
twenty years 1887-1906 there were 2,107 deaths from explosions of fire-
damp or coal-dust in the coal-mines of the United Kingdom, or an average
of 105 deaths per annum. From 17.15 it follows that this should be the
square of the standard deviation of simple sampling, or the standard
deviation itself approximately 10-3. But the square of the actual
standard deviation (the standard error) is 7,178, or its value 84*7, the
numbers of deaths ranging between 14 (in 1903) and 317 (in 1894). This
large standard deviation, to judge from the figures, is partly, though not
wholly, due to a general tendency to decrease in the numbers of deaths
from explosions in spite of a large increase in the number of persons
employed ; but even if we ignore this, the magnitude of the standard
deviation can be accounted for by a very small value of the correlation r,
expressive of the fact that if an explosion is sufficiently serious to be fatal
to one individual, it will probably be fatal to others also. For if o-q denote
the standard deviation of simple sampling, a the standard deviation of
sampling given by equation (17.11), we have —

’’ “^-l)cro2

Whence, from the above data, taking the numbers of persons employed
underground at a rough average of 560,000,

7,073

560,000x105

+0-00012

17.38 Summarising the preceding paragraphs, 17.30-17.37, we see that
if the chances p and q differ for the various populations, districts, years,
materials, or whatever they may be from which the samples are drawn,
the standard deviation observed (the standard error) will be greater than the
standard deviation of simple sampling, as calculated from the average values
of the chances ; if the average chances are the same for each population
from which a sample is drawn, but vary from individual to individual or
from one sub-class to another within the population, the standard deviation
observed (the standard error) will be less than the standard deviation of
simple sampling as calculated from the mean values of the chances ; finally,
if p and q are constant, but the events are no longer independent, the
observed standard deviation (the standard error) will be greater or less
than the simplest theoretical value according as the correlation between
the results of the single events is positive or negative. These conclusions

THE SAMPLING OF ATTRIBUTES

407

further emphasise the need for caution in the use of standard errors. If we
find that the standard deviation in some case of sampling exceeds the
standard deviation of simple sampling, two interpretations are possible :
either that p and q are different in the various populations from which
samples have been drawn (i.e. that the variations are more or less signifi-
cant), or that the results of the events are positively correlated inter se.
If the actual standard deviation fall short of the standard deviation of
simple sampling two interpretations are again possible : either that the
chances p and q vary for different individuals or sub-Uasses in each popula-
tion, while approximately constant from one population to another, or
that the results of the events are negatively correlated inter se. Even if
the actual standard deviation approaches closely to the standard deviation
of simple sampling, it is only a conjectural and not a necessary inference
that all the conditions of simple sampling '' are fulfilled. Possibly, for
example, there may be a positive correlation r between the results of the
different events, masked by a variation of the chances p and q in sub-
classes of each population.

An alternative approach

17.39 The results of this chaptei have been studied from a rather different
point of view by a continental school of statisticians, among whose names
those of Lexis and Charlier are prominent.

Lexis considers a number of samples of n individuals in which the
proportions of successes observed are pi, p 2 > • • • Pn> 2 ,nd sets himself
to investigate the nature of the population from which they were drawn —
whether it is homogeneous and the samples may be regarded as obtained
by simple sampling, whether it varies in time or place so that the samples
are not simple, and so on. He takes p to be the mean of the observed
values Px * pN> writes —

r = 0-67449.

y n

He then defines

R = 0-67449^5^^£^

where the summation extends over aU values oi pi . , . p^^, and writes

17.40 Now, if the sampling is simple we may, in large samples, take
the mean p to be an estimate of the true value, and r to be an estimate of
the probable error of simple sampling of Also, we may take the quantity
i? to be an estimate of the probable error of p (see 21.7).

Hence, for large samples, R is approximately equal to r, and 0=^-1,
This case, which is w^hat we have called simple sampling. Lexis calls
** normal dispersion.’*

4o8

THEORY OF STATISTICS

17.41 On the other hand, if the population is not constant while the
samples are drawn, or if they come from different parts of a patchy popula-
tion, we get the case discussed in 17.30. R is no longer an estimate of the
probable error of a constant p, but may be split into two parts, one due to
the sampling fluctuations of the observed values of p round the mean value,
the other due to the variations of the true values round that mean. R will
therefore be greater than r, as may be seen from equation (17.8), and
^>1. This case Lexis calls '"supernormal dispersion.*’

17.42 Similarly, in the case discussed in 17.32 we get R less than r,
and hence Q<1. This case Lexis calls "subnormal dispersion,” and
speaks of the data which give nse to it as " constrained ” (gebundene).

The quantity Q is analogous to a quantity which we shall consider
at some length in Chapter 20 in discussing the significance of the deviations
of observed frequencies from theoretical expectation.

SUMMARY

1. Under simple sampling conditions, the proportion of successes in a
sample may be taken as an estimate of the proportion of successes in the
parent population.

2. If p is the proportion of successes in the population, the standard error
of simple sampling of the number of successes is given by

cr = 's/npq

and of the proportion of successes by

3. The probability that an observed number of successes deviates from
the expected number by more than three times the standard error is very
small This fact enables us to set limits to the range within which the
observed frequency lies when we know the theoretical frequency.

A. Fc" ivirge samples, the observed frequency of successes may be used
to calculate the standard error, and this fact enables us to set limits to
file range wiibin which the theoretical frequency lies when vre know the
obser/'d {nqc'':w}\

5. lor samples, if the chance of success varies from sample to

sample but remains constant within a sample, the standard error of the
number of successes is given by

==. npQq^+n{n--l)ap^
ana of the proportion of successes by

a j ^ ^

THE SAMPLING OF ATTRIBUTES 4O9

where pQ is the mean of the varying chance of success, is the standard
deviation of p, and n is the number of individuals in each sample

If n is large and Sq is the standard deviation calculated from the mean
pQ, this last equation is approximately

6. If the chance of success varies between the individuals of a sample
but does not vary as between the different samples,

= np^q^—na^^

^2 Po^o

n n

7. If the chance of success remains constant for each member of each
sample, but the events are not independent,

0-2 = npq{l-\-r{n—\)}

where r is the mean of the correlations between the results of the events.

EXERCISES

17.1 Compare the actual with the theoretical mean and standard deviation
for the following record of 6,500 throws of 12 dice, 4, 5 or 6 being reckoned
as a '' success —

Successes

Frequency

Successes

Frequency

1,351

844

103

391

302

117

711

1,231

1,411

Total

6,500

17.2 (Quetelet, '' Lettres . . . sur la thdorie des probabilitds.")

Bails were drawn from a bag containing equal numbers of black and white
bails, each ball being returned before drawing another. The records were
then grouped by counting the number of black balls in consecutive 2*s,
3's, 4's, 5's, etc. The following are the distributions so derived for

410

THEORY OF STATISTICS

grouping by 5's, 6’s, and 7's. Compare actual with theoretical means
and standard deviations.

Successes

(a) Grouping
by fives

(5) Grouping
by sixes

{c) Grouping
by sevens

125

277

166

104

224

192

151

136

166

148

—

Total

819

683

585

17.3 The proportion of successes in the data of Exercise 17.1 is 0*5097.
Find the standard deviation of the proportion with the given number of
throws, and state whether you would regard the excess of successes as
probably significant of bias in the dice.

17.4 In the 4,096 drawings on which Exercise 17.2 is based 2,030 balls
were black and 2,066 white. Is this divergence probably significant of
bias ?

17.5 (Data from Report I, Evolution Committee of the Royal Society,
page 17.) In breeding certain stocks, 408 hairy and 126 glabrous plants
were obtained. If the expectation is one-fourth glabrous, is the divergence
significant, or might it have occurred as a fluctuation of sampling ?

17.6 400 eggs are taken at random from a large consignment, and 50 are
found to be bad. Estimate the percentage of bad eggs in the consignment
and assign limits within which the percentage probably lies.

17.7 In a certain association table (data from Exercise 2.5) the following
frequencies were obtained—

{AB) ^ 309, {A^) = 214, {aB) = 132, (a^) == 119

Can the association of the table have arisen as a fluctuation of simple
sampling, the true association being zero ? .

17.8 The sex ratio at birth is sometimes given by the ratio of male to
female births, instead of the proportion of male to total births. If Z is
the ratio, J.e. Z^p jq, show that the standard error of Z is approximately

(l+Z)^ n being large, so that deviations are small compared with

the mean.

17.9 In a random sample of 500 persons from town A, 200 are found
to be consumers of cheese. In a sample of 400 from town B, 200 are also

THE SAMPLING OF ATTRIBUTES

4II

found to be consumers of cheese. Discuss the question whether the
data reveal a significant difference between A and B so far as the propor-
tion of cheese-consumers is concerned.

17.10 In a newspaper article of 1,600 words in English 36 per cent of
the words are found to be of Anglo-Saxon origin. Assuming that simple
sampling conditions hold, estimate the proportion of Anglo-Saxon words
in the WTiter's vocabulary and assign limits to that proportion.

Suggest possible causes which might break down the three conditions
for simple sampling.

17.11 If a senes of random samples of different sizes is taken from the
same material, show that the standard deviation of the observed propor-
tions of successes in such sets is s, where

and E is the harmonic mean of the numbers in the samples.

17.12 Apply the result of the previous exercise to the following data
(A. D. Darbishire, Biometnka, voL 3, page 30), giving percentages to the
nearest unit of albinos obtained in 121 litters from hybrids of Japanese
waltzing mice by albinos, crossed inter se —

Percentage

Frequency

Percentage

Frequency

100

Calculate the actual standard deviation and compare it with the result
given by the formula of the previous exercise. The expected proportion
of albinos is 25 per cent, and the sizes of the litters are given in Example
5.5, page 121

17.13 In a case of mice-breeding (see reference above) the harmonic mean
number in a litter was 4 * 735, and the expected proportion of albinos 50
per cent. Find the standard deviation of simple sampling for the propor-
tion of albinos in a litter, and state whether the actual standard deviation
(21*63 per cent) probably indicates any real variation, or not.

17.14 If for one half of n events the chance of success is p and the chance
of failure q, whilst for the other half the chance of success is q and the
chance of failure p, what is the standard deviation of the number of
successes, the events being all independent ?

41 ^

THEORY OF STATISTICS

17.15 Corresponding to the case of equation (17.8) show that if the values
of are small so that the binomial tends to the Poisson limit with parameter
M, the variance of the numbeis of successes observed is given by

where M is the mean value of M and is the standard deviation.

17.16 Similarly, corresponding to equation (17.10), show that

s^ = M

so that the usual equation for the standard error holds notwithstanding
departures from simple sampling of the type here considered, (cf.
equation (17.3)).

17.17 The following are the deaths from smallpox during the twenty
years 1882-1901 in England and Wales —

1882

1,317

1892

431

957

1,457

2,234

820

2,827

223

275

541

506

1,026

253

174

1900

1901

356

The death-rate from smallpox being very small, the rule of 17.15 may
be applied to estimate the standard deviation of simple sampling. Assum-
ing that the excess of the actual standard deviation over this can be
entirely accounted for by a correlation between the results of exposure
to risk of the individuals composing the population, estimate r. The
mean population during the period may be taken in round numbers as
29 millions.

CHAPTER EIGHTEEN

THE SAMPLING OF VARIABLES

LARGE SAMPLES

Sampling of variables

18.1 We are now able to proceed from the sampling of attributes to
the sampling of variables. Whereas in the last chapter we were interested
in the question whether a member of a sample did or did not exhibit a
particular attribute, we now have to study individuals which may take any
of the values of a variable. It will no longer be possible, therefore, for us
to classify each member of a sample under one of two heads, success or
failure ; in general the values of the variate given by different trials will
be spread over a range, which may be unlimited, limited by practical
considerations, as in the case of height in human beings, or limited by
theoretical considerations, as in the case of the correlation coefficient,
which cannot lie outside the range +1 to —1.

18.2 To give concreteness to our discussions we shall occasionally find
it useful to consider the sampling of variables as a kind of ticket sampling.
We may picture our population as made up of tickets, each bearing a
recorded value of some variable X. Sampling may then be imagined to
consist of the drawing of tickets and the noting of the values of X which
they bear. In the great majority of cases with which we shall deal, X
may have any value over a continuous range, and the ticket population
is to be conceived as being actually or practically infinite.

18.3 As in the case of attributes, our principal objects in studying
these samples will be {a) to compare observation with expectation and to
see how far deviations of one from the other can be attributed to fluctua-
tions of sampling ; (5) to estimate from samples some characteristic of the
parent population, such as the mean of a variate ; and (c) to gauge the
reliability of our estimates.

In order to grasp satisfactorily the ideas and assumptions upon which
work of this kind is based, it is necessary to develop some theoretical
considerations which have already been touched upon in the last chapter.
This we now proceed to do.

Sampling distributions

18.4 If we take a number of samples from a population and calculate

413

414

THEORY OF STATISTICS

some function,^ such as the mean or the standard deviation, of each sample,
we shall in general get a series of different values, one for each sample. If
the number of samples is at all large, these values may be grouped in a
frequency distribution ; and as the number of samples becomes larger,
this distribution will approach the '' ideal '' form of a continuous curve.
Such a distribution is called a sampling dtstnbution.

18.5 As an illustration, consider the population of 8,585 men, classified
according to height, of Table 4.7, page 82. In Chapter 16 we showed
how to draw a random sample of 10 individuals from this population,
and for one sample we calculated the mean. The following table shows
the 100 values of the sample mean obtained by taking 100 such samples
arranged in the form of a frequency table —

TABLE 18.1. — Frequency distribution of means of samples of 10 from the population
of the last column of Table 4.7 page 82

Value of mean in
sample (inches)
less inch

Number of samples with
specified values of
the mean

64*4-

64-8-.

—

65-2-~

65*6-

66*0-

66*4-

66*8-

67-2-

67*6-

68*0-

68*4-

Total

100

This distribution is not very regular, owing to the smallness of the total
frequency.

18,6 As a second illustration we take some data obtained with random
sampling numbers from a bivariate normal population with correlation
+0-9. 500 samples of 10 were taken and the correlation coefficient
of each sample worked out. The frequency distribution of the 500 values
was as follows (data adapted from P. R. Rider, Distribution of Correla-
tion Coefficient in Small Samples," Biometrika, voL 24, 1932, page 382)—

^ Quantities such as means, standard deviations, moments, correlation coefficients
and so forth will be referred to generically as parameters.'* It is the modem practice
to reserve this word for a population value and to denote the corresponding sample
value by the word * 'Statistic." Thus a sample-mean is a statistic which forms the
estimate of a population-mean, the parameter.

THE SAMPLING OF VARIABLES 415

TABLE 18.2. — Frequency distribution of correlation coefQcients in samples of 10 from

a normal population

Value of r m sample

Frequency

0- 1-0-2

0-2-0-3

0-3-0-4

0-4-0-5

0- 5-0-6

0-6-0-7

0-7-0-8

102

0-8-0-9

178

0*9-1 -0

131

Total

500

Here the distribution is more regular, the number of samples being five
times as large. In general we expect that as the number of samples
increases, the distribution will tend more and more to a continuous curve.

Use of the sampling distribution

18.7 Let us suppose that we are given the sampling distribution of a
statistic, and that the frequency (y) may be represented in terms of
the variate (x) by a continuous curve,

y ==/W

The frequency with which a given value of the statistic occurs in
a large number of samples will be represented by the ordinate of the
curve at the point whose abscissa is Xq, We have had an example of
this in the normal curve.

The number of samples which give a value of x greater than Xq will be
represented by the area to the right of the ordinate at Xq ; the number
giving a value less than Xq will be represented by the remaining area to
the left.

Hence, the chance that any sample chosen at random from all possible
samples will give a value of x greater than Xq is given by the area to the
right of the ordinate at divided by the total area of the curve, which
represents the total number of samples ; and the chance that the sample
will give a value of less than % is given by the area to the left of the
ordinate of Xq divided by the total area.

Similarly, the chance that a sample would give a value of a; lying
between, say, Xj and x^ is the area lying between the ordinates at the points
Xi and X 2 divided by the total area.

4i6

THEORY OF STATISTICS

18.8 In 8.21 we referred to the fact that areas could be expressed in
the notation of the integral calculus. In fact, we may write the area
of the curve between and as

[^fix)dx

and hence we may express P, the probability that a sample will give a
value between Xy^ and x^, as

where we assume the extreme hmits to be ± oo as in the normal curve.
In particular, the probability that the sample will give a value of a; greater
than Xq is given by

P =

As a rule, we can choose our units so that the area of the curve is unity.
This simplifies the above expressions ; for the denominator, being equal
to unity, may be omitted.

18.9 Now let us suppose that, knowing the form of the sampling distribu-
tion and hence being able to calculate P for any given Xq, we take a
sample and find that it gives a very low value of P. We are then faced
with three possibilities : either a very improbable event has occurred ;
or the assumptions on which we obtained the sampling distribution were
incorrect ; or there is something wrong with our sampling technique.
Which of these explanations we adopt is to some extent a matter of choice,
but if we have tested our sampling, or on other grounds have no reason
to suspect it, we shall, as a rule, be led to query the hypotheses on which
the sampling distribution was obtained.

This, in effect, is what we did in the previous chapter. It so happens
that in the simple sampling of attributes we know that the exact form
of the sampling distribution is N{q+p)^, where p is the chance of success.
Without examining this distribution too closely we can say that only a
very small part of it lies outside the range ± 30 . Hence, if we find a
sample giving a value outside the range iiSVn^q, we suspect the hypothesis
on which the distribution was based ; and this, unless we prefer to suppose
that our sampling was not in fact simple, leads us to suspect the value of
p, which completely determines the sampling distribution.

18^10 In the previous chapter we regarded the probability of a sample
giving a value differing by more than 3a from the mean value as so remote

THE SAMPLING OF VARIABLES

417

that in every case we should be justified m looking for some definite
cause of the discrepancy. This is only a conventional range, based upon
the empirical fact that in most single-humped populations it includes
nearly all the members ; but it is a convenient one to take and we shall
use it again below. For certain purposes, however, we might be prepared
to use a narrower range which, though not giving such a small probability
that a sample lay outside it, yet indicated considerable improbability in
the divergence of observation from expectation, and enabled us to criticise
the validity of our hypotheses with some degree of assurance. We give
one or two examples below'.

18.11 In practice nearly aU the sampling distributions we have to
consider are based on simple sampling. It is therefore convenient to
speak briefly of a " sampling distribution,*' meaning thereby a sampling
distribution obtained under simple (and random) conditions.

Example 18.1. — The sampling distribution of a statistic is a normal
population with mean 9 units and standard deviation 2 units. What is
the probability that a sample will give a value of the statistic greater
than 12 units ?

Here the value 12 is three units, i.e. l-5a, to the right of the mean.
The required probability is therefore the area of the normal curve to the
right of an ordinate 1 • 5a to the right of the mean, divided by the total
area of the curve.

This ratio can be obtained at once from Table 2 of the Appendix.
We see, in fact, that the greater fraction of the area of the curve corre-

spending to -==1 -5 is 0*9332. The smaller fraction is therefore 0*0668,
a

which gives us the required probability.

Example 18.2. — If the sampling distribution of a statistic is normal,
with zero mean and standard deviation a, what is the value of the sta-
tistic such that the chances are 99 to 1 against a sample giving a value
in excess of that value ?

We have to find x such that the area of the curve to the right of the
ordinate at is 0*01. or the area to the left 0*99

From Appendix Table 2 —

If -—2*32, greater fraction of area =0*9838

and if- =2-33 „ „ =-0-9901

Hence, to the nearest second place of decimals the required value is 2*33cr.

Example 18.3. — It very frequently happens in sampling inquiries
that we are interested in the probability that a sample value exceeds a
given value x^ in absolute value, i.e. that it is greater than or less than

4i8

THEORY OF STATISTICS

—Xq We can ascertain this probability without much trouble from the
ordinary table of areas of the normal curve if the distribution is normal.
Consider, for instance, the data of Example 18.1. Here we found the
probability that a sample would give a value greater than l*5a. If we
want the probability that it would give a value greater than l*5cr in
absolute value, we have —

P = Area to right of ordinate at 1 ‘So
+ Area to left of ordinate at —1 -Scr

Since the curve is symmetrical, the two areas in question are equal, and

P =2(1 -0-9332)

= 0-1336

18.12 To apply the results of 18.7 to 18.11 in practice for the purpose
of discussing the population from which the samples came, we require to
know two things : (a) What is the relation between the sampling dis-
tribution and the parent distribution, and what is the form, at least
approximately, of the sampling distribution of a given statistic from a
given population ?

18.13 If the sampling is to be of much use in enabling us to estimate
the value of a parameter in the parent, we should expect most of our
estimates to be somewhere near the mark, and only comparatively few to
be very far from the true value of the quantity estimated ; and further, we
expect that, in general, the further the estimates are from the truth the
fewer there will be of them.

To put this more formally, we expect that the sampling distribution
will have a peak somewhere close to the value of the parameter which
corresponds to the true value in the parent. If it does not, the distribution
is probably biased and our samples are likely to be misleading.

The first desideratum in our sampling is, therefore, that it shall not lead
to a biased distribution. We have seen in Chapter 16 the difficulties of
eliminating bias in the sampling process itself. Where, therefore, the more
practical considerations alluded to in that chapter impose no limitation,
we must use unbiased sampling ; and this means that our sampling must
be random. In this connection it must be remembered that we cannot
. judge from the samples themselves whether the sampling is random or not,
though we may suspect it. Separate tests, or the use of some accredited
method, are to be recommended where practicable.

18.14 Knowledge of the form of the sampling distribution of. a statistic,
even of an approximate kind, is by no means easy to secure. We saw that
in' the case of the simple sampling of attributes it was possible to deduce
the sampling distribution in an exact form. We are not always in this

THE SAMPLING OF VARIABLES 4I9

fortunate position here — in fact, rarely so. The principal difficulties
are —

[a) The form of the parent population frequently is unknown.

(h) Even if the form of the parent is known, certain of its constants may
be unknown ; for instance, we may know that a population is normal but
be ignorant of its mean and standard deviation.

(c) If the parent is completely known, the form of the sampling dis-
tribution can be deduced theoretically in certain circumstances, and in
particular if the sampling is simple ; but in practice the mathematical
problems which arise usually are very complex, and even if they are
tractable may be of no use owing to the enormous arithmetical labour
involved in expressing a solution in serviceable form.

18.15 If the samples are small these difficulties are formidable, even
for simple sampling. With large samples, however, we are able to make
certain legitimate approximations and assumptions which greatly simplify
the problem. For the lest of this chapter and in the next we shall be
concerned solely with large samples.

Simple sampling of variables

18.16 We shall also be thinking mainly in terms of simple sampling
(17.3). It is unnecessary to recapitulate here the discussion of simple
sampling which we gave in the previous chapter. The assumptions which
we considered in 17.19 to 17.24 apply mutatis mutandis to the simple
sampling of variables.

{a) We assume that we are drawing from precisely the same record
during the whole of the sampling ; if we picture our parent population
as a card population, the chance of drawing a card with any given value
X is the same for each sample.

{h) We assume not only that we are drawing from the same record
throughout, but that each of our cards at each drawing may be regarded
quite strictly as drawn from the same record (or from identically similar
records) : e.g. if our card record is contained in a series of bundles, we must
not make it a practice to take the first card from bundle number 1, the
second card from bundle number 2, and so on, or else the chance of drawing
a card with a given value of X, or a value within assigned limits, may not
be the same for each individual card at each drawing,

(c) We assume that the drawing of each card is entirely independent
of that of every other, so that the value of X recorded on card 1, at each
drawing, is uncorrelated with the value of X recorded on card 2, 3, 4, and
so on. It is for this reason that we spoke of the record, in 18.2, as contain-
ing a practically infinite number of cards, for otherwise the successive
drawings at each sampling would not be independent : if the bag contains
ten tickets only, bearing the numbers 1 to 10, and we draw the card bearing
1 , the average of the following cards drawn will be higher than the mean of

420

THEORY OF STATISTICS

all cards drawn ; if, on the other hand, we draw the 10, the average pf the
following cards will be lower than the mean of all cards — i.e. there will be
a negative correlation between the number on the card taken at any one
drawing and the card taken at any other drawing. Without making the
number of cards in the bag indefinitely large, we can, as already pointed out
for the case of attributes, ehmmate this correlation by replacing each card
before drawing the next.

Approximations in the theory of large samples

18.17 We can now consider the approximations which are possible in
the theory of large samples.

In the first place, since we have supposed bias to be eliminated, the
sample values of a statistic will be grouped about the true value, and
if the samples are large, will differ by comparatively small quantities
from that value. Hence, we may take a sample value as an estimate
of the true value. That is to say, if we have a large sample (which may
consist of a number of samples run together), we may calculate the para-
meter from it precisely as we should proceed if we were calculating the
parameter for the population as a whole, and take that value as our
estimate. Thus, the mean of the sample may be taken as an estimate
of the mean of the population.

18.18 This rule is not quite so obvious as it appears. Suppose, for
example, that we are estimating the standard deviation of a population.
In accordance with the previous paragraph we should take the standard
deviation of the sample. But in calculating this quantity we should have
to use deviations, not from the true mean, but from the mean in the sample,
which may differ from the true mean and to that extent affect the value
of the estimate. We shall, in fact, see later that if x^, jCg . . . are the
values in the sample and x their mean, there are reasons for preferring

the estimate to the estimate s^ = -'E(X'~x)^ for the

variance. If n is large, however, the difference is unimportant ; we can
ignore it until we come to deal with small samples.

18.19 Secondly, as in the case of attributes, we can use these estimates
in calculating the constants of the sampling distnbution, since they
differ only by small quantities from the real values. We saw, for instance,
that we were justified in taking the value of p in a large sample in
calculating the standard deviation y'npq of the sampling distribution.
We shall find that the standard deviation of the sampling distribution of
the mean of samples from a normal population involves the standard
deviation of the parent ; and in this case we can evaluate that quantity
by using the standard deviation of the sample in place of the unknown
standard deviation of the parent.

THE SAMPLING OF VARIABLES 421

18.20 Finally, it is a very remarkable fact that the sampling distributions
of many statistics, obtained under simple sampling conditions, tend
for large samples to a single-humped form either exactly or very closely
normal. The evidence for this statement is partly theoretical, partly
experimental. It mav be shown that, for simple samples from a normil
population, the sampling distributions of most statistics are exactly
normal for large samples — some, in fact, are normal for small samples.
Following up this work, a number of experiments has been carried out on
populations which are not normal ; and it appears that the parent can
deviate quite markedly from the normal form without affecting the nor-
mality of the sampling distribution to any great extent provided, as before,
that the samples are large.

In most of our work we shall not require to assume that the sampling
distribution is normal. It will be sufficient to assume that a range of 3a
on each side of the mean includes the major portion of the distribution,
and we can confidently take this to be so unless the parent exhibits very
marked skewness.

18.21 It will now be apparent that the difficulties we specified in 18.14
have to a great extent been met. Provided that we laiow the parent
distribution to be not unduly skew, we need not know its exact form ;
and the sampling distribution can be represented satisfactorily, if not
exactly specified, by a mean and standard deviation which may be
estimated from the data of the sample.

Standard error

18.22 As in the last chapter, we shall refer to the standard deviation
of the sampling distribution as the standard error. In most cases we
shall be dealing with simple samphng distributions, but it is convenient
to use the term in this wider sense, although the word '' error is not
altogether appropriate in some instances. In general, as we have seen,
we are justified in taking a range of ± 3 times the standard error as deter-
mining limits outside which the value of the parameter given by a sample
probably does not lie. We can therefore use the standard error, as we
have already used it for attributes, to gauge the precision of an estimate
or to permit a judgment being made of the divergence between expected
and observed values.

In the remainder of this chapter, and in the next, we shall therefore
be concerned mainly in finding expressions for the standard errors of
the various parameters which we have to estimate. Their use we shall
illustrate in examples as we go along. In certain cases we shall also
consider the effect of a breakdown in the conditions of simple sampling.

Standard of error of a quantile, quartile and median

18.23 Let us first of all consider the case of quantiles, which is intimately
related to that of attributes.

422

THEORY OF STATISTICS

Consider the distribution of a variate X in an indefinitely large sample.
(This is not necessarily the same as the distribution in the parent, owing
to the possible presence of bias ; but if bias is excluded, and the sampling
is simple, it is the same as the parent form.)

Let Xj, be a value of X such that pN values of A" in this distribution
lie above it and qN below it. Thus, if the sampling is unbiased, p =
would give us the upper decile m the indefinitely large sample, p~^ the
median, and so on.

A sample of n will contain various values of X, Let the proportion
of values above X^ be ; a^nd let e be the adjustment to be made in
Xj, so that the proportion of values of X above X^ + e is p. The values
S and e may be regarded as sampling fluctuations.

Considering now the sample of n, we have that

Hence,

the proportion of values above Xj, = pJ^S

ti f> Xjf~\~6 P

$ = proportion of values between Xj, and Xj, + 6

Now if n be large, the proportion of values between X^ and Xj,+e in
the sample will, to a close approximation, be the proportion of values
between those quantities in the distribution of an indefinitely large
sample. Consider then this distribution and let the standard deviation
of X in it be o. If we take the distribution as drawn to scale with unit
standard deviation and unit area, the proportion of values between

and X^+e is the area of the curve between ordinates at the points —

and

Now if w be large, e will be small, for the value of a parameter in the
sample of n will lie close to the value in the indefinitely large sample.

X^+e

Hence the area between — and — — is approximately rectangular, and

X e

if we call the — ordinate y«, the area will be y«X —
a . ^ cr

Hence,

THE SAMPLING OF VARIABLES

423

Now ^ is the deviation of the observed proportion from the value p ;
and from our study of attributes we know that the observed proportion

p d will centre round the mean p with standard deviation a

V n

Hence d centres round zero mean with standard deviation

Ip<i

Since

e bears a constant ratio — to S, it follows that e will be distributed about
zero mean with standard deviation

Vvar [x^) =

yjy ^

(18.1)

18.24 If the distribution in an indefinitely large sample be normal,
we can take the values of from the tables of the ordinate of the normal
curve (Appendix Table 1). From tables carried to further places of
decimals v^e have, for the various values of p which correspond to the
deciles,

Median

Value of yj>

. 0-3989423

Deciles 4 and 6

. 0-3863425

„ 3 and 7

. 0-3476926

„ 2 and 8

. 0-2799619

„ 1 and 9

. 0-1754983

Quartiles

. 0-3177766

Inserting these values of in equation (18.1), we have the following
values for the standard errors of the median, deciles, etc. —

Standard error is
cr/V« multiplied by

Median .... 1 -25331

Deciles 4 and 6 ... 1 *26804

„ 3and7 . . . 1-31800

„ 2and8 . . . 1-42877

Iand9 . . . 1-70942

Quartiles .... 1-36263

It will be seen that the influence of fluctuations of sampling on the
several quantiles increases as we depart from the median : the standard
error of the quartiles is nearly one-tenth greater than that of the median,
and the standard error of the first or ninth decile more than one-third
greater.

18.25 Consider further the influence of the form of the frequency-
distribution on the standard error of the median, as this is an important
form of average. For a distribution with a given number of observations

4^4

THEORY OF STATISTICS

and a given standard deviation the standard error vanes inversely as yp.
Hence for a distribution in which yp is small, for example a U-shaped
distribution, the standard error of the median will be relatively high, and
it will, in so far, be an undesirable form of average to employ. On the
other hand, in the case of a distribution w^hich has a high peak in the
centre, so as to exhibit a value of yp large compared with the standard
deviation, the standard error of the median will be relatively low. We
can create such a peaked distnbution by superposing a normal curve
with a small standard deviation on a normal curve with the same mean
and a relatively large standard deviation. To give some idea of the
reduction in the standard error of the median that may be effected by a
moderate change in the form of the distribution, let us find for what
ratio of the standard deviations of two such curves, having the same area,
the standard error of the median reduces to a jVn, where a is of course
the standard deviation of the compound distribution.

Let Cj, 02 be the standard deviations of the two distributions, and let
there be n /2 observations in each. Then

. (18,2)

On the other hand, the value of yp is

l2V27rai 2V2n<j.^

Hence, the standard error of the median is

JV'-

V 27 T cr^cr,
n cr^+<

+^2

(18.4) is equal to c jVn if

2'\/7r<JiCT2

and writing that is if

(18.3)

(18.4)

{l+p)Vl+p2 _ 1

2 V 7 TP

p4^2pH(2-47r)p2+2p + l =0

This equation may be reduced to a quadratic and solved by taking

~ as a new variable. The roots found give p — 2 *2360 ... or
P

0-4472 . . , , the one root being merely the reciprocal of the other. The

THE SAMPLING OF VARIABLES

425

standard error of the median will therefore hea jVn, in such a compound
distribution, if the standard deviation of the one normal curve is, in round
numbers, about 2J times that of the other. If the ratio be greater, the
standard error of the median will be less than (jjVn. The distribution
for which the standard error of the median is exactly equal to cj Vn is
shown in fig. 18.1 ; it will be seen that it is by no means a very striking
form of distribution ; at a hasty glance it might almost be taken as normal.
In the case of distributions of a form more or less similar to that shown,
3t is evident that we cannot at all safely estimate by eye alone the relative
standard error of the median as compared with a/Vn.

18.26 In the case of a grouped frequency-distribution in which the
number of observations is large enough to give a fairly smooth distribution,
we can use a alternative form which does not involve a knowledge of the
standard deviation of the distribution in a very large sample. In fact, in
such a case the sample itself is large enough to give us a satisfactory
approximation to the distribution in an indefinitely large sample. Let fp

be the frequency per class- interval at the given percentile — simple inter-
polation will give us the value with quite sufficient accuracy for practical

purposes, and if the figures
run irregularly they may
be smoothed. Let o be
the value of the stan-
dard deviation expressed in
class-intervals, and let n
be the number of obser-
vations as before. Then,
since yp is the ordinate of
the frequency-distribution
when drawn with unit
standard deviation and unit
area, we must have

yp == -fp

But this gives at once for
the standard error expressed

Fig. 18.1

tn terms of the class-interval
as unit

(18.5)

Example 18.4. — Consider the data of Table 4.7, page 82, giving the
distribution of 8,585 men according to height. Let us take these data to
be a sample from the population of men in the United Kingdom at that

426

THEORY OF STAlISTICS

time. The number of observations is 8,585, and the standard deviation
2*57 in., the distnbution being approximateiy normal a 027737,
and, multiplying by the factor 1 -253 , . . given m the table m 18.24, this
gives 0*0348 as the standaid error of the median, on the assumption of
normality of the distribution.

Using the direct method of equation (18.5), \^e hnd the median to be
67*47 (5.20), which is very nearly at the centre of the interval with a
frequency 1,329. Taking this as being, with sufficient accuracy for our
present purpose, the frequency per interval at the median, the standard
error is

^V8585
^ 1329

0-0349

As we should expect, the value is practically the same as that obtained
from the value of the standard deviation on the assumption of normality.

Three times the standard error is 0* 1047, and we accordingly conclude
that the median m the population lies within about 0* 1 inch of 67*47, the
sample value, provided that the sampling is simple.

Example 18.5. — Let us find the standard error of the first and ninth
deciles as another illustration. On the assumption that the distribution
is normal, these standard eriors are the same, and equal to 0*027737
X 1 *70942=0*0474. Using the direct method, \\e find by simple inter-
polation the approximate frequencies per interval at the first and ninth
deciles respectively to be 590 and 570, giving standard errors of 0*0471
and 0*0488, mean 0*0479, slightly m excess of that found on the assump-
tion that the frequency is given by the normal curve. The student should
notice that the class-interval is, in this case, identical with the unit of
measurement, and consequently the answer given equation (18.5) does
not require to be multiplied by the magnitude of the interval.

Correlation between errors of quantiles

18.27 In finding the standard error of the difference between two quantiles
in the same distnbution, the student must be careful to note that the
errors in two such quantiles are not independent. Consider the two
quantiles for which the values of p and q are p^^ p^ q^, respectively,
the first named being the lower of the two quantiles. These two quantiles
divide the whole area of the frequency curve into three parts, tlie areas
of which are proportional to p^. Further, since the

errors in the first quantile are directly proportional to the errors in ^ 1 ,
and the errors in the second quantile are directly proportional but of
opposite sign to the errors m p^, the correlation between errois in the two
quantiles will be the same as the correlation between errors in q^ and p^,
but of opposite sign. But if there be a deficiency of observations below the
lower quantile, producing an error 8^ in the missing observations will

THE SAMPLING OF VARIABLES

427

tend to be spread over the two other sections of the curve in proportion to
their respective areas, ^ and will therefore tend to produce an error

in If then, r be the correlation between errors m and p^, and Cr
the respective standard errors, we have —

Or, inserting the values of the standard errors.

The correlation between the quantiles is the same in magnitude but
opposite in sign , it is obviously positive, and consequently

Correlation between errors
in two quantiles

If the two quantiles approach very close together, and q^ pn and ^2
become sensibly equal to one another, and the correlation becomes unity,
as we should expect. An alternative derivation is suggested in 19.3.

Standard error of semi-interquartile rang^

18.28 Let us apply the above value of the correlation between quantiles
to find the standard error of the semi-interquartile range for the normal
curve. Inserting Hence the standard

error of the interquartile range is, applying the ordinary formula for the
standard deviation of a difference, times the standard error of

either quartile, or the standard error of the smwnterquartile range
1 1 a/3 times the standard error of a quartile. Taking the value of the
standard error of a quartile from the table in 18.24, we have, finally.

Standard error of the semi- ]
interquartile range in a h =
normal distribution )

Of course the standard deviation of the interquartile, or semi-inter-
quartile, range can readily be worked out in any particular case, using
equation (18.5) and the value of the correlation given above ; it is best to
work out such standard errors from first principles, applying the usual
formula for the standard deviation of the difference of two correlated
variables (14*2).

1 This statement is, perhaps, not obviously true, and the assumption which it represents
IS not a necessary condition for the validity of equation (18.6) . The alternative approach
of 19.3 avoids using it.

428

THEORY Ol STATISTICS

18.29 If there is any failure of the conditions of simple sampling, the
formulae of the preceding sections cease, of course, to hold good. We
need not, however, enter again into a discussion of the effect of removing
the several restrictions, for the effect on the standard error of p was con-
sidered m detail in Chapter 17, and the standard error of any quantile is
directly proportional to the standard error of p.

Standard error of the arithmetic mean

18.30 Let us now determine the standard error of the arithmetic mean.

Suppose we note separately at each drawing the value recorded on the

first, second, third . . . and ^th card of our sample. The standard deviation
of the values on each separate card will tend in the long run to be the
same, and indentical with the standard deviation a of a; in an indefinitely
large sample, drawn under the same conditions. Further, the value
recorded on each card is (as we assume) uncorrelated with that on every
other. The standard deviation of the sum of the values recorded on the
n cards is therefore y/no, and the standard deviation of the mean of the
sample is consequently 1 jni\\ of this , or,

. , . (18.8)
y n

This is a most important and frequently cited formula, and the student
should note that it has been obtained without any reference to the size of
the sample or to the form of the frequency-distribution. It is therefore
of perfectly general application, if cr be known. We can verify it against
our formula for the standard deviation of sampling in the case of attributes.
The standard deviation of the number of successes m a sample of m observa-
tions is \/^pq ' the standard deviation of the total number of successes
in n samples of m observations each is therefore \/nmpq * dividing by n we
have the standard deviation of the mean number of successes in the n
samples, viz. yfmpq j^/n, agreeing with equation (18.8).

Example 18.6. — In the height distribution considered in Examples 18.4
and 18.5 we found that a / y^S== 0*0277 approximately. This is then
the standard error of the mean of the distribution.

If we regard the data as a simple sample from the population of men in
the United Kingdom, we may take the mean, i.e. 67*46 inches, as an
estimate of the mean in the population. Three times the standard error
is very small, 0*083 inch, and we can therefore locate the mean in the
population with considerable accuracy.

The standard error in this case, however, gives a misleading idea as
to the accuracy attained in determining the average stature in the United
Kingdom ; the sample was not chosen under conditions which gave every
individual an equal chance of being chosen.

THE SAMPLING OF VARIABLES

429

Comparison of the standard errors of the median and the mean

18.31 For a normal curve the standard error of the mean is to the
standard error of the median approximately as 100 to 125 (cl 18,24),
and in general the standard errors of the two stand in a somewhat similar
ratio for a distribution not differing largely from the normal form. For
the distribution of statures used as an illustration in Example 18.4, the
standard error of the median was found to be 0-0349 : the standard error
of the mean is only 0*0277. The distribution being very approximately
normal, the ratio of the two standard errors, viz. 1-26, assumes almost
exactly the theoretical magnitude.

As such cases as these seem on the whole to be more common and
typical, we stated in 5.23 that the mean is ^n general less affected than
the median by errors of sampling. At the same time we also indicated the
exceptional cases in which the median might be the more stable — cases in
which the mean might, for example, be affected considerably by small
groups of widely outlying observations, or m which the frequency-distribu-
tion assumed a form resembling fig. 18.1, but even more exaggerated
as regards the height of the central peak '' and the relative length of
the " tails.” Such distributions are not uncommon in some economic
statistics, and the}^ might be expected to characterise some forms of ex-
perimental error. If, in these cases, the greater stability of the median
is sufficiently marked to outweigh its disadvantages in other respects, the
median may be the better form of average to use. Fig. 18.1 rfepresents
a distribution in which the standard errors of the mean and of the median
are the same. Further, in some experimental cases it is conceivable that
the median may be less affected by definite experimental errors, the average
of which does not tend to be zero, than is the mean — this is, of course, a
point quite distinct from that of errors of sampling.

Means of two samples

18.32 When we have two samples from some record which exhibit
different means, a very common question which we wish to ask is . Can
the difference be accounted for by sampling fluctuations, i.e. can the two
samples have come from the same population ^

If the two samples are independent and come from the same population
(Under simple conditions, evidently “the standard error of the difference
of their means, is given by

If an observed difference exceed three times the value of given by
this formula, it can hardly be ascribed to fluctuations of sampling. If, in
a practical case, the value of q is not known a priori, we must substitute
an observed value, and it would seem natural to take as this value the
standard deviation in the two samples thrown together. If, however, the

430

THEORY OF STATISTICS

standard deviations of the two samples themselves differ more than can
be accounted for on the basis of fluctuations of sampling alone (see below,
19.14), we evidently cannot assume that both samples have been drawn
from the same record . the one sample must have been drawn from a
record or a population exhibiting a greater standard deviation than the
other. If two samples be drawn quite independently from different
populations, indefinitely large samples from which exhibit the standard
deviations (Ji and Oo, the standard error of the diffeience of their means
will be given by

This is, indeed, the formula usually employed for testing the significance
of the difference between two means in any case ; seeing that the standard
error of the mean depends on the standard deviation only, and not on the
mean, of the distribution, we can inquire whether the two populations
from which samples have been drawn differ in mean apart from any difference
in dispersion,

18.33 If two quite independent samples be drawn from the same popula-
tion, but instead of comparing the mean of the one with the mean of the
other we compare the mean of the first with the mean of both
samples together, the use of (18.9) or (18.10) is not justified, for errors
in the mean of the one sample are correlated with errors in the mean
of the two together. Following precisely the lines of the similar problem
in 17.29, we find that this correlation is and hence

(18.11)

Effect on standard error of mean of breakdown of conditions for simple
sampling

18.34 Let us consider briefly the effect on the standard error of the
mean if the conditions of simple sampling as laid down in 18.16 cease
to apply.

If we do not draw from the same record ail the time, but first draw a
series of samples from one record, then another series from another record
with a somewhat different mean and standard deviation, and so on, or if
we draw the successive samples from essentially different parts of the same
record, the standard error will be greatly increased.

For suppose we draw samples from the first record, for which the
standard deviation (in an indefinitely large sample) is and the mean
differs by d-^ from the mean of all the records together (as ascertained by
large samples in numbers proportionate to those now taken), samples
from the second record, for which the standard deviation is cr^, and the

THE. SAMPLING OF VARIABLES

431

mean difers by from the mean of all the records together, and so on.
Then for the samples drawn from the first record the standard error of the
mean will be crj but the distribution will centre round a value dif ering
by from the mean for all the records together ; and so on for the samples
drawn from the other records. Hence, if be the standard error of the
mean in all the records taken together, N the total number of samples,

But the standard deviation Gq for all the records together is given by

Hence, writing 'L{kd^) =

■ +

n — l

(18.12)

This equation corresponds precisely to equation (17,8), page 401. The
standard error of the mean, if our samples are drawn from different records
or from essentially different parts of the entire record may be increased
indefinitely as compared with the value it would have in the case of
simple sampling. If, for example, we take the statures of samples of
n men in a number of different districts of England, and the standard
deviation of all the statures observed is Gq, the standard deviation of the
means for the different districts will not be Gq/V n, but will have some
greater value, dependent on the real variation in mean stature from
district to district.

18.35 If we are drawing from the same record throughout, but always
draw the first card from one part of that record, the second card from
another part, and so on, and these parts differ more or less, the standard
error of the mean will be decreased. For if, in large samples drawn from
the subsidiary parts of the record from which the several cards are taken,
the standard deviations are a^, Og, . . . cr„, and the means differ by
d^, . . . d^ from the mean for a large sample from the entire record,
we have —

ao»=^S(a=)+is(<i=)

Hence,

. (18.13)

432

THEORY OF STATISTIC^

The last equation again corresponds precisely with that given for the
same departure from the rules of simple sampling in the case of attributes
(equation (17.10), page 403). If, to vary our previous illustration, we
had measured the statures of men in each of n different districts, and
then proceeded to form a set of samples by taking one man from each
district for the first sample, one man from each district for the second
sample, and so on, the standard deviation of the means of the samples
so formed would be appreciably less than the standard error of simple
sampling cr^lVn. As a limiting case, it is evident that if the men in each
district were all of precisely the same stature, the means of all the samples
so compounded would be identical ; in such a case, in fact, cro=s^, and
consequently To give another illustration, if the cards from which

we were drawing samples had been arranged in order of the magnitude of
X recorded on each, we would get a much more stable sample by drawing
one card from each successive wth part of the record than by taking the
sample according to our previous rules — e.g. shaking them up in a bag
and taking out cards blindfold, or using some equivalent process.

The result is perhaps of some practical interest. It shows that, if we
are actually taking samples from a large area, different districts of which
exhibit markedly different means for the variable under consideration, and
are limited to a sample of n observations, if we break up the whole area
into n sub- districts, each as homogeneous as possible, and take a contribu-
tion to the sample from each, we will obtain a more stable mean by this
orderly procedure than will be given, for the same number of observations,
by any process of selecting the districts from which samples shall be taken
by chance. There may, however, be a greater risk cf biased error. These
conclusions seem in accord with common sense. We consider this subject
further in Chapter 23.

18*36 Finally, suppose that, while our conditions (a) and {b) of 18.16
hold good, the magnitude of the variable recorded on one card drawn
is no longer independent of the magnitude recorded on another card,
e.g. that if the first card drawn at any sampling bears a high value, the next
and following cards of the same sample are likely to bear high values also.
In these circumstances, if denote the correlation between the values
on the first and second cards, and so on,

= '^+ 2^(^12 + ^ 13 + * • +^ 23 + • * •)

There are n(w — 1)/2 correlations; and if, therefore, r is the arithmetic
mean of them all, we may write —

■ ■ • ( 18 . 14 )

As the means and standard deviations of are all identical.

THE SAMPLING OF VARIABLES

433

r may more simply be regarded as the correlation coefficient for a table
formed by taking all possible pairs of the n values in every sample. If this
correlation be positive, the standard error of the mean will be increased,
and for a given value of r the increase will be the greater, the greater the
size of the samples. If r be negative, on the other hand, the standard error
will be diminished. Equation (18.14) corresponds precisely to equation
(17.12), page 405.

As was pointed out in 17.35, the case when r is positive covers the
case discussed in 18.34 ; for if we draw successive samples from different
records, such a positive correlation is at once introduced, although the
drawings of the several cards at each sampling are quite independent of
one another. Similarly, the case discussed in 18.35 is covered by the case
of negative correlation, for if each card is always drawn from a separate
and distinct part of the record, the correlation between any two x's will
on the average be negative ; if some one card be always drawn from a part
of the record containing low values of the variable, the others must on an
average be drawn from parts containing relatively high values. It is as
well, however, to keep the three cases distinct, since a positive or negative
correlation may arise for reasons quite different from those considered in
18.34 and 18.35.

SUMMARY

1. A knowledge of the sampling distribution of a statistic enables us
to ascertain the probability that a given sample will exhibit a value of the
statistic between specified limits.

2. The sampling distribution of many statistics tends to the normal
form, or at least a single- humped form, for large values of n, the number in
the sample, if the sampling is simple.

3. This fact enables us to take a range of ±3 times the standard error
as providing limits within which a sample value of the statistic will
probably lie ; with the fuither assumption of normality of the sampling
distribution we can determine the probability that a sample value will lie
within any specified limits.

4. In a large sample the values of statistics in the sample may be
taken to be estimates of the values in the population, if the sample is
simple. Further, these values may be used instead of the values in the
population in calculating the standard errors of the statistics.

5. The standard error of the median of a normal distribution is given by

s.e. = 1*25331 —

where a is the standard deviation in an indefinitely large sample and n
is the number in the sample.

434

T E 0 R Y OF STATISTICS

6. With the same notation the standard error of the arithmetic mean is

s e. = ~ 7 =-
V n

whatever the form of the distiibution.

7. If a series of samples of n is drawn from different populations or from
different parts of a non~homogeneous population,

where cr^ is the standard error of the mean, is the standard deviation
in all the samples taken together, and is the standard deviation of
means of indefinitely large samples about the mean of all samples.

8. If samples are drawn so that each member comes from a different
section of a non«homogeneous population.

where or^, <Jq and are defined as before.

9. If there is a correlation between the results of the drawing of succes-
sive individuals,

where is the standard error of the mean, cr the standard deviation in
an indefinitely large sample, and r is the mean correlation between the
results of pairs of individuals.

EXERCISES

18.1 If the sampling distribution of a statistic is normal, find the
probability that a sample value will differ from the central value by more
than twice the probable error.

18.2 In the height distribution of the United Kingdom given in Table
4,7, page 82, assumed to be normal, with mean 67*46 inches and standard
deviation 2*57 inches, find the probability that an individual chosen in
the same way as the members of the distribution will be between 5 and 6
feet in height.

18.8 For the data of the last column of Exercise 4.6, page 100, find the
standard error of the median (154*7 lbs.) and the standard errors of the
two quartiles (142*5 lbs, and 168*4 lbs.)

18.4 For the same distribution find the standard error of the semi-inter-
quartile range.

THE SAMPLING OF VARIABLES

435

18.5 The standard deviation of the same distribution is 2T3 lbs. Find
the standard error of the mean and compare it with the standard en'or
of the median (Exercise 18.3).

18.6 Taking the values of the median and the quartiles of the maniage
distribution of Table 4.8, page 84, from Example 7.8, page lOO, find their
standard errors.

18.7 In the same distribution the mean is 29 ‘4 years and the standard
deviation 8 years, approximately. Find the standard error of the mean
and compare it with that of the median.

18.8 For the same distribution find the standard error of the quartiles,
assuming it to be normal with mean 29*4 years and standard deviation
8 years, and compare your results with those obtained in Exercise 18.6.

18.9 Find the standard error of the 27th percentile of the normal dis-
tribution.

18.10 (Imaginary data.) A random sample of 1,000 men from the North
of England shows their mean wage to be £2 7s. per week, with a standard
deviation of £l 8s. A sample of 1,500 men from the South of England
gives a mean wage of £2 9s. per week, with a standard deviation of £2.
Discuss the suggestion that the mean rate of wages varies as between
the two regions.

18.1 1 Two populations have the same mean but the standard deviation of
one is twice that of the other. Show that in samples of 500 from each
drawn under simple random conditions the difference of the means will in
aU probability not exceed 0'3a, where a is the smaller standard deviation ;
and assuming the distribution of the difference of means to be normal,
find the probability that it exceeds half that amount,

18.12 A random sample of 1,000 farms in a certain year gives an average
yield of wheat of 2,000 lbs. per acre, with a standard deviation of 192 lbs.
A random sample of 1,000 farms in the following year gives an average
yield of 2,100 lbs. per acre, with a standard deviation of 224 lbs. Show
that these data are inconsistent with the hypothesis that the average yields
in the country as a whole were the same in the two years.

Would you modify this conclusion if the farms in the second sample
were the same as those in the first ?

18.13 Find the mean and median of the U-shaped distribution of Table
4.14, page 96, and compare their standard errors. (For the purpose of
this exercise the median frequency may be found by simple interpolation,
but this gives a value on the high side.)

18.14 The mean of a certain normal distribution is equal to the standard
error of the mean of samples of 100 from that distribution. Find the
probability that the mean of a sample of 25 from the distribution will be
negative.

436

THEORY OF STATISTICS

18.15 If it costs a shilling to draw one member of a sample, how much
would it cost, in sampling from a population with mean 100 and standard
deviation 10, to take sufficient members to ensure that the mean of the
sample in all probability would be within 0-01 per cent of the true value ?
Find the extra cost necessary to double the precision.

18.16 Consider the data of Table 4 7, page 82, giving the distribution
of men by height in each of the four countries which then formed part
of the United Kingdom. The means and standard deviations of the four
distributions are given in Exercise 5.1, page 122 and Exercise 6.1, page 148.

What is the standard error of the mean of a sample which consists of
400 men, 100 chosen at random from each of the four countries ?

CHAPTER NINETEEN

THE SAMPLING OF VARIABLES

LARGE SAMPLES, CONTINUED

The problem

19.1 We have just considered the standard errors of the most important
measures of location, the median and the mean, and of certain measures
of dispersion, the quantiles and the semi-interquartile range. We now
proceed to discuss the standard errors of other important parameters,
including the standard deviation, moments and correlation coefficients.
All that we have said in regard to sampling distributions generally in

18.1 to 18.22 applies equally well to this chapter ; and we shall throughout
the following sections be thinking of simple sampling unless we state
explicitly to the contrary.

Standard errors of moments^

19.2 The data from which we calculate the moments are arranged into

a certain number of groups. Suppose there are m such groups, and
that the expected frequencies falling into them are • •

where yi+y 2 + • • • ^ being the number in the

sample. The expected frequencies are, as shown below, proportional to
the frequencies in the various groups of the parent population.

Let us in the first place recapitulate some of our earlier work by finding
the stanaard error of one of the frequencies, say y,, due to fluctuations of
sampling.

The probability that an individual chosen from the population falls
into the sth group is'^ The probability that it does not is 1 — ~ For w
individuals the distribution of frequencies is given by the binomial

with an expected value and a standard deviation

1 The student whose main interest lies in the practical application of the results of
this chapter may prefer to omit paragraphs 19.2 to 19.8.

437

438

THEORY OF STATISTICS

Now, if the sample is large, we can take the observed frequency in the
5 th group in calculating the standard error of the frequency of that group.
Taking this observed frequency as our estimate of its standard error,
is given by

vav (y,) = . . . (19.1)

This in another form, is our familiar result for the sampling of attributes.

19.3 We may now find the correlation between errors in and errors
in another group-frequency, say y*. It is evident that such a correlation
will exist, for if jtq falls below its expected value, some other frequencies
must be increased.

Consider the variance of + y<. We have, from (14.3), page 327,
var (ys4-yi) == var y,+varyf+2 cov (y^, y^) . . fl9.2)

Substituting for the variances from (19.1) with the similar expression

var {y,+yt) = (:y's+>'j)(^l
we find, after a little rearrangement

2 cov
whence

cov {ys,yi) ==

(19.3)

This is a more general case of the correlation between quantiles which
we considered in 18.27, For the correlation between y^ and y^ we have,
on dividing (19.3) by the standard deviations —

1 -

iL.]^

19.4 By definition the ^th moment about an arbitrary point is where

X being the variate measured from the arbitrary point. We write a
deviation in a quantity or y^ as Sja^ or ^y^ as the case may be. (The
symbol d is not to be regarded as a number multiplying /ir or y^ but
as part of the single quantity d/1' or dVs-)

Squaring both sides,

wWP* == • • • +•*«%«)-

= i:{x.^{Sy,)^}+2-L'{x,^x,Wyt)

THE SAMPLING OF VARIABLES

439

where E' denotes summation over all values of s and t except those for
which s ^ i.

This equation holds for any one sample, and we have to sum it for all
samples. Carrying out this summation first (in which s and i are fixed),
and substituting from equations (19.1) and (19.3) on the right-hand side,
we have —

var /Zg' = = 2

-2S‘

= S(a./»v<,) (xfyt)

Hence,

Vvar jjLq = a »

. (19.4)

Example 19.1. — Let us find the standard error of the first moment,
or mean h.

We have, from (19.4) —

-V-

Now is the second moment about the mean, i e. is

Hence,

'7^ 0*

which is the result we have already found in 18,30.

Correlation between errors in the fth and fth moments, both about the
same fixed point
19.5 As in 19.4 we have —

nSiig' =

nSfi/ = 'L{x,^8y^

Multiplying,

+S'{ {x,W+x;x,^){dy,8yt ) }

and summing for all samples,

cov (/t'„ /i'r) ==S(«:/+'’ vax y,)+I.'[[x,^x/-\-x;xt9){cov

440

THEORY OF STATISTICS

On substitution for var and cov (v„ v^) from (19.1) and (19 v'^), the right-
hand side reduces to aiid hence,

cov (/^; //;)

(19 5 )

Standard error of the moments about the mean

19.6 In 19.4 and 19.5 we have considered moments about a fixed point.
In practice we have to deal more usually with moments about the mean
oj ihe sample. Since this mean is itself subiect to sampling fluctuations,
the standard errors of moments about the mean will not in general be the
same as those about a fixed point

lih IS the mean we have, by definition,

W<i

== + r

where T is written generally for an expression involving and higher
pow’ers of h.

Now let h vary to vary to y^+^Vs, and //. vary to

We have-- ' - a

= ^^‘'(yASy.)} -q{h-{Sh)Z{rtHys-^ Sy,)}+T
Subtracting the equation for n/i^,

= ~ —nqdkS/i'^^^ + U

where U will involve h and higher powers. We may neglect the term in
as being small compared with the remaining terms. Squaring
and summing for all samples,

var /^^=var var h-2qfi'^.^ cov {h, /^) + U

Substituting for var //' etc. from (19 4) and (19.5),

var u jj

Now put A=0. U vanishes and the moments become moments about
the mean and may therefore be written without dashes. Hence,

V'^var

==%-->/•

gVa/tr-1-
n

-29/^a-i/^'cr+:

(19.6)

Correlation between two moments both measured about the mean
19.7 In a similar way it may be shown that

cov /^g/^r d" 4-1

^ n

We omit the algebra for the sake of brevity.

(19.7)

THE SAMPLING OF VARIABLES

441

Correlation between errors in a moment about a fixed point and in a
moment about the mean

19.8 Let us first of all find the correlation between deviations in a
group-frequency yt and the moment ju^' about a fixed point. We havf' :

Hence,

=xH^ytV+'^'(x,^dy,dyt)

the summation S' being taken over all values of s except s—/.
Hence, summing for all samples,

Hence,

n cov (/ 4 ', Vt)

cov (4.

(19.8)

Similarly, for the product-sum of deviations in yq and the moment /(^
about the mean, we have —

cov (/«„ V,)

74- /*'

-fterms in h and higher powers
Putting the right-hand side reduces to

For the product-sum of errors in and

-f U

wdiere U, as before, denotes an expression involving h and higher powers.
Hence,

Summing for all deviations,

cov (/i', cov (y/ 5 , /a;)} — cov (A, y,)} + U

and substituting from (19.8) and (19.9) the right-hand side becomes

-f 1/^r-i , jj

n n

442

THEORY OF STATISTICS

Use of Sheppard’s corrections in evaluating standard errors.

19,9 Theoretically, Sheppard’s corrections for grouping are not to be
used in evaluating the moments which enter into the general equations for
standard errors obtained in the previous sections. For, as the corrected
values differ from the uncorrected values only by constants depending on
the width of the interval, the sampling deviations of corrected and un-
corrected moments are equal, and hence so are their standard errors. But
the standard errors of uncorrected moments are given by the equations we
have obtained in the foregoing section, and hence those equations are
applicable to corrected moments provided that the uncorrected values are
used in them.

In practice, however, it seems to make very little difference which
moments we use, unless the sample is very large indeed. But as the
uncorrected values have to be obtained before the corrected values can be
calculated, and are therefore usually available, it is as well to use the
uncorrected values wherever possible.

Standard error of the variance

19.10 Armed with the general results of the foregoing sections, we can
discuss the standard errors of a large class of parameters.

From equation (19.6), putting we have, since /^i==0,

Vvar

= /A4

(19.11)

which gives the standard error of the variance
If the parent population is normal,

A2 = fU == (8,23)

and hence,

(19.12)

Standard error of the standard deviation
19Jl1 If Wg is the variance, we have—

fti = a*

THE SAMPLING OF VARIABLES

443

Hence,

= {(J+Say

= a2+2a<^a+((5>'o‘)^

Neglecting S<j^ in comparison with Sa,

Sji2 = 2<jd(j

Squaring and summing for all samples,

var ^2 = cr 2 = 4a2 var a
/^2

Hence,

Vvar a- = aa a„ — —

” 2<J y 4fi^n

If the parent distribution is normal this reduces to

. (19.13)

y'var a = = — ^ .... (19.14)

V2m

19.12 The form of equation (19.14) has been widely used for the standard
error of q without due regard to the nature of the parent population,
and the student should guard against this mistake.

We have, in fact, from (19.13) —

V /{.
'v/2«

-1

How far cr„ can be taken to be the value (19.14) therefore depends on
how close the factor ^1 is to unity, i.e. depends on the kurtosis

of the parent distribution.

The following table shows the value of this factor for various values
oifii—

8
9

0- 7071

1 - 0000
1-2247
1-4142
1-5811
1-7321

1- 8708

2 - 0000

THEORY OF STATISTICS

444

It thus appears that if the population is leptokurtic the real standard
error is greater than that given by the assumption of normality, and may
be twice as great or even more. If the population is platykurtic the real
standard error is less than the '' normal " value.

is small, the factor ^ 1 +^^^y is approximately

This difers from unity by more than 5 per cent if y ?2 is less than 2-8 or
more than 3 • 2. Hence, values of outside the range 2 • 8 to 3 • 2 (and

they are more common than not m practice) wnll give an error of more than
5 per cent if the population is assumed to be normal.

Example 19.2. — For the height distribution of Table 4.7, page 82, we
have found that a=2*57 inches, ^=8585. The population may be taken
to be normal, for from the sample is 3* 149 (Example 7.9, page 164) and

2*57

hence the standard error of cr= -- — = 0*02 approximately.

V2X8585

Hence, we may say that the s.d. in the population almost certainly lies
in the range 2 *57^:0 *06, assuming that the sampling is simple.

Example 19.3. — The distribution of Australian marriages of Table 4.8,
page 84, has uncorrected moments fi^ and ^^ 4 , in class-intervals, as follows —

Hence,

/Zg = 7*0570

/i 4 — 408*7382 (Example 7.2, page 157.)
C7 = V/Zg = 2*6565

The standard error of a =

V 4/i^n

408 *7382 -(7 *0570) 2
4x7*0570x301,785'

= 0 • 00649 class-intervals

As we should expect from such a large sample, the standard error is
very small, and we conclude that the standard deviation of the parent
lies in the range 2 *6565 ±0*01 95.

It may be pointed out that if we take these data as a sample of
Australian marriages in general, we may be violating the conditions of
simple sampling, for the distribution most likely changes from year to
year.

Example 19.4. — In the previous example we worked throughout with
uncorrected values. The corrected moments (Example 7 . 4 , page 159)
are —

CK 6*9736
/t 4 == 405*2389

THE SAMPLING OF VARIABLES

445

We then have, for the corrected value of a,

a = ^6-9736
= 2-641

But the standard error of a is 0-00649 as in the previous example, for we
must use the uncorrected values m calculating it.

As a matter of fact, if we had used the corrected values we should
have found the value 0 * 00654 — a practically negligible difference even for a
sample of this size.

Finally, let us compare this value with that given by the assumption
of normality. We have —

a =

V60i,570

= 0 • 00342 class-intervals

i.e. only about half the true value. This is in accordance with the result
of Example 7.6, for is over 8.

Comparative effects of sampling fluctuations and corrections for grouping
19.13 Writing temporarily ctj^ for the uncorrected value of the variance
and 02 ^ for the corrected value, we have —

If the class-interval is chosen so as to make the number of intervals d

then 6a, would be about dh and — about j- Hence

" Oj d

2d^

For instance, if d is 20, the coriected value is about 0-375 per cent less
than the uncorrected value.

Now, for a normal population,

or, since — is small

446

THEORY OF STATISTICS

and if n is, say, 1 ,000, the standard error is ==0 • 0224a =2 • 24 per cent

of a. Thus Sheppard's correction amounts to no more than about one-
sixth of the standard error, and to make it gives an almost misleading
idea of precision in most practical cases.

It was for this reason that we recommended (6.12 and 9.29) that the
Sheppard corrections should not be applied if the total frequency is less
than 1,000. On the other hand, in Examples 19.3 and 19.4 the correction
is large compared with the standard error and can reasonably be made,
Giving to the largeness of the sample.

Comparison of standard deviations of two samples

19,14 As in 18,32, where we considered the comparison of the means
of two samples, if the samples are independent and come from the same
population the standard error of the difference of their standard deviations
is given by

€

4/t2

- i-l

. (19.15)

where are the numbers in the samples, or, if the population be

normal,

€

i!/l 1

2 W j ft 2

. (19.16)

If the two samples are drawn from different populations with constants
/i2, and Fa, F4, the standard error of the difference of the standard

deviations is given by

. (19.17)

2% 2«2

. (19.18)

if the population be normal.

Again, if the standard deviation of one sample is compared with the
standard deviation of the two samples when pooled, the standard error of
the difierence is, if the distribution be normal,

. (19.19)

These results can be used to test the significance of differences between
standard deviations precisely as the equations of 18.32 and 18.33 were
used to test the significance of differences between means.

THE SAMPLING OF VARIABLES

447

Standard error of third and fourth moments about the mean
19.15 From equation (19.6), putting ^ = 3,

-6/44/(2+9/<j®

. (19.20)

If the distribution is normal,

= 15a«, /I4 = 3a^ /ig = 0, /ig =

Hence,

rrS

Similarly, from equation (19.6), putting ^=4,

//^8 — -“ 8 /^ 5/^3 + 1 6 /^ 2 ^ 3 ^

< =y

If the distribution is normal, /^ 5 = 0 .

Hence,

rr^

<^/^4 = :^ Vi 05-9

'96

““■V-

. (19.21)

(19.22)

. (19.23)

Example 19.5. — For the height distribution of Table 4.7 we have
(Example 7.1, page 153) —

/^2 (uncorrected) == 6*6168

/ig (uncorrected) = —0*2078
/i 4 (uncorrected) = 137*6892

and from Example 7.3, page 159 —

/ig (corrected) — 6*5335

/ig (corrected) = —0*2078
/i 4 (corrected) = 134*4100

We did not calculate higher moments, and hence cannot use equations
(19.20) and (19.22) with these data. The distribution is, however,
approximately normal. Hence, from (19.21),

= 0*45 approximately

The value of /ig cannot therefore be judged significantly different from
zero, which is what we should expect, for we have assumed the population
to be normal.

448

THEORY OF STATISTICS

From (19.23) we have —

— 4-63 approximately

These are calculated from the uncorrected value of a. We may infer
that /^4 (corrected) lies within the range 134 -41 ±13*89. The Sheppard
correction is only 3 • 28, and is submerged in the possible sampling deviation,
even for a sample of 8585. What we have said m 19.13 applies, in fact,
a fortiori to the higher moments.

19.16 It will be evident that the standard errors of moments of high
order are very large , for the moments increase rapidly, and the standard
error of the moment of order q depends on the moment of order 2q, For
example, in the normal distribution, for q=6, /i 2 ?==l 0,3950"^ ^ and will

be of the order whereas /46==.15a®. Unless, therefore, n is at least

400, the range will be greater than the value of and hence we
cannot locate the value of in the population with any exactness. Our
approximations, in fact, break down if the deviations are large.

The large sampling errors of moments of high orders prevent the use
of moments higher than the fourth in most practical problems.

Correlation between errors in mean and standard deviation

19.17 From equation (19.10), putting ^==1, r~2, and remembering that
/ii=-0, we have —

—a r

Hence, if errors in the mean and variance, and hence in the

mean and s.d., are uncorrelated. In particular, we have the important
result that errors in the mean and s.d. in a normal population are un-
correlated. In actual fact they are independent, even for small samples,
but we shall have to state this result without proof.

Standard error of the coefficient of variation

19.18 The coefficient of variation V is defined as

V =1^
h

_ lOOV/^g

THE SAMPLING OF VARIABLES

449

Hence,

r-i 5r =

h+8h

IOOVA 2 /, , 5^-'

/'a/ \

1 2/t2jl A]

Neglecting quantities small compared with and Sh, this becomes

Hence,

51' _ S/I 2 8h
'V "" T

/iJt'

Summing for all samples we have-

cji - _ var /^2 , var h cov (fi^, h)
r*^ 4/^2^ li^ ^io]i

If the distribution is normal-

and cov (//g, h) = 0 ^ 19 . 17 ).
Hence,

i a z!

2n h-H
2n\ 10^

Hence,

V2ny 10 ^

(19,24)

In many practical cases the second term differs little fiom unity and

= will give a sufficiently precise result.

450

THEORY OF STATISTICS

Standard error of and /?»

19.19 The standard errors of and p^ can be deduced in a similar
manner.

In fact,

Pi-

A+% —

which, after some reduction, gives

2/<3a/<3 S/Ta^

Squaring and summing for all samples —

4;/-2

(/^«-/'3®-6A'4Aa-9A8®)

4/^3^ .. , 9/^3*

12/13*

6 var/i 3 +^ var/io-“-^f- cov (/ij, /i,)
r2 /^2 r 2

= — X

9/13^

In terms of /?j, Pz A (see page 159, footnote, for definition of the
higher /?’s),

varA-§{4A4-24/?.,+36+9AA3-12/?3+35A} ■ (19.25)

/t'

Similarly,

vary^a = h/?3-4;^3y54+4y?3*-/?3*+16/?3A-8y?3+16A} • (19.26)

The labour of evaluating these quantities may be obviated by the use
of tables given in Tables for Statisticians and Biometricians, Part L

19,20 There is here one important point to be noted. In equation
(19.24), if (jy=^0. Similarly, in equation (19.25), if cr^3^=0.

It might be thought from this that if in a large sample we find in the one
case that V =0 (and hence that a==0), or in the other case that the distri-
bution is symmetrical, then F==0 or /?i=0 in the population. This is not
necessarily true.

V will vanish only if all members of the sample give the same value
of the variate. If the sample is large, it will be evident that if there is
any variation in the parent it must be small : but it is not impossible
that members should exist showing deviations from the observed value.
The explanation is to be found in the terms which we have neglected

THE SAMPLING OF VARIABLES

451

m our approximations. These, though in general small compared with
the terms retained, may be important if the terms retained themselves
vanish. Futhermore, our assumption that the sample value is the same
as the parent value may be unjustified if both are very small compared
with their difference. Equations such as (19.24) and (19.25) must, there-
fore, be treated carefully in the neighbourhood of values which cause them
to vanish.

19.21 From the foregoing work the student will have no difficulty in
accepting the statement that it is possible to calculate the standard
error of any quantity which is expressible as a function of the moments.
Such a standard error would, however, be applicable only to a value
which had actually been calculated from the moments, and not arrived
at by some other means. We shall not pursue the subject further m this
book, but we may point out that the standard errors of certain quantities,
such as an approximation to the Pearson measure of skewness (7.12), have
been tabulated in Tables for SfaHstictans and Btomstnaans for different
values of and The same tables also contain some results of interest
in connection with the sampling distributions of range.

We now turn to the parameters of multivariate universes, the correla-
tion coefficients, regression coefficients, and some of the measures of
association.

Standard error of the correlation coefficient

19.22 For samples from a normal population the standard error of the
correlation coefficient is given by

A proof of this result would take us beyond the scope of the present
work. It has to be used with reserve for values of the correlation near
to unity, since the distribution in such a case is markedly skew unless
the sample is very large, say, at least 500. When there is any doubt it
is better to use an alternative test given in 21.33,

The formula applies also to partial correlations.

19.23 Formula (19.27) is sometimes used to estimate the precision of
correlation coefficients obtained by the use of the product-moment formula
without reference to the nature of the population. This practice is
hardly to be commended, although sometimes there is nothing better
to do. It is, however, possible to generalise the procedure of sections
19.2 to 19,8 to the bivariate case, and it may be shown that

r® 4 /e.o/^02 hiHo ^ ^

(For the definition of the bivariate moments, see footnote, page 222).

452

THEORY OF STATISTICS

In addition, if the regression is linear, denoting the of the two
variates considered separately by >

( 1 — ' 1

which reduces to (19.27) if the kurtosis is zero.

If the distribution is not normal and r is not small, the difference between
the values given by (19.27) and (19.29) may be considerable ; but it may
be noticed that the value given by (19.27) is less than that given by (19.29)
if the distribution is platykurtic for both variates, and greater if the
distribution is leptokurtic for both variates.

19.24 In particular, it may be shown that for a 2x2 table m which
the frequencies are (AB), (AJ3), {uB) and (a/?), the standard error of the
correlation coefficient calculated by the product-moment method on the
assumption that the frequencies are concentrated at points is given by

V(A)(a){Bm

{A){a)

[(g)-(A}]q)

(sm Jj

(19.30)

19.25 The standard error of tetrachoric r, as calculated in the manner
of 11.32, is given by very complicated expressions which we do not
reproduce. The coefficient is very sensitive to departures of the parent from
normality, and no satisfactory test of significance seems to be known
Example 19.6. — In the data of Table 9.3, page 202, we found that
the correlation between the stature of the father and the stature of the
son was 0*51. Regarding these data as a sample of 1078 from the popula-
tion of fathers and sons, we have —

c. . . X l--(0-51)^

Standard error of r = — 7=- = — 7=rr-

^/n Vl078

= 0 • 023 approximately

Hence, if the sampling was simple, the correlation in the population
most probably lies within 0*44 and 0-58. It is thus undoubtedly real.

Example 19.7. — In considering data from 14,416 cows, J. F. Tocher
found a negative correlation of 0*0796 between yield of milk per week and
percentage of butter fat. Is this significant, i.e., could it have arisen from
an uncorrelated population by sampling fluctuations ?

If r=0,

THE SAMPLING OF VARIABLES 453

The correlation observed is ten times this, and small though it is,
could not have arisen from sampling fluctuations.

In this example we may reiterate the caution to be observed in inferring
from the sample anything about the population (cows in Scotland) as
a whole The records were, in fact, taken by the Scottish Milk Records
Association from constituent associations at various years between 1908
and 1923. The conditions of simple sampling may, therefore, have been
violated both in regard to time and in regard to place.

Standard error of the coefficient of regression

19.26 The standard error of the coefficient of regression from a normal
population is given by

CTg Vn Og V«

. (19.31)

This again applies to a regression coefficient of any order, total or
partial, i.e., in terms of our general notation, k denoting any collection of
secondary subscripts other than 1 or 2,

Standard error of \
for a normal distribution j cig &

The correlation ratio and coefficient of multiple correlation
19.27 It has been shown that the sampling distributions of the correlation
ratio and the multiple correlation coefficient from normal populations
do fiot tend to the normal form for large samples, although they do give
single-humped distributions. The use of a standard error in such cases
must be made with great caution, and it is probably better to apply
one of the tests of significance which we shall consider later in connection
with the theory of small samples. The formula usually given for the
standard error of the correlation ratio is an approximate one —

. (19.32)

J9 28 Somewhat similar remarks apply to the coefficient ^= 71 ^
which, as we saw in 11.8, may be used to test the linearity of regression.
The use of a standard error for f in an attempt to gauge the significance of
a departure from linearity has been subjected to very damaging criticism.

Example 19.8. — Consider the data of Example 12.2, page 293 (relation
between pauperism, age of population and number of population).

We found —

jtj, = 0*325 a;2+1-383%-0-383:s?4

Taking this to be given by a random sample from a normal population,
is the value 0*325 significant ?

454

THEORY OF STATISTICS

We have —

__ 234 ^1,234^ ^ 34

^2 34 V' « <^2 134 «

_ 22 -sVl -0-457"

~ 32-1^32

= 0-11

The coefficient &12.34 is therefore significant.

In this example the number in the sample is not as large as one might
wish and the standard error is probably underestimated ; but if any
doubt exists it is possible to make more definite tests by the methods of
Chapter 21.

Standard error of coefficient of association

19.29 We may refer briefly to the quantities treated in Chapters 2 and 3,
in considering the association of attributes.

The coefficient of association, Q, defined in 2.15, has a standard error
given by

i -(?2 / r ~i 1 1

2 y

. (19.33)

This quantity is not infinite, as might at first sight appear, if one of
the cell frequencies vanishes, because in that case 1 — <2^ also vanishes; in
fact, in such an event a^—O.

Standard error of the coefficient of mean-square contingency

19.30 The determination of the standard error of the coefficient of
mean-square contingency is a matter of considerable mathematical com-
plexity, and even when approximations are employed, leads to expressions
which are tedious to calculate in practice. For a detailed discussion we
must refer the student to the original memoirs (K. Pearson, Biometrika,
1913, 9, 22 and T. Kondo, Biometrika, 1929, 21, 376).

Spearman’s rank correlation coefficient

19.31 Unlike most of the parameters we have been considering, the
distribution of Spearman's rank correlation coefficient is discontinuous,
and to that extent resembles the binomial. Very little is known about the
distribution except in the important case when the correlation in the
population is zero. The other cases are sometimes treated by assuming a
normal continuous distribution in the parent and working from ranks to
grades and thence to the product-moment coefficient of correlation by
the equations (11.21) and (11.23) of 11.29; but this procedure is not
to be recommended.

THE SAMPLING OF VARIABLES

455

The case when the correlation in the population is zero, i.e., when all
possible permutations of the ranks occur with equal frequency, has to some
extent been investigated. It was shown by Student ” in 1907 that the
standard deviation of Spearman's rank correlation coefficient is given
by the simple equation

ap- .... (19.34)

This cannot be taken to be a standard error in the ordinary way,
because the distribution is not normal for small samples. It has also
been shown that the distribution tends to normality as n increases, but
for low values of n the normal distribution gives an unsatisfactory approxi-
mation. For values of n greater than 8 the significance of an observed p
can be tested in the ^-distribution (see below, 21,25) by entering the tables
with t=pV{n—2)lV[\—p'^) and

The rank correlation coefficient t

19.32 For the coefficient r more information is available. Kendall
{Advanced Theory of Statistics, VoL 1, chapter 16) has given the actual
distribution up to and including ^^ = 10 in the case wffiere all possible
rankings occur equally frequently, and has shown that the distribution
tends to normality more lapidly than that of p. For values greater than
w = 10 the distribution can be assumed to be normal with a standard
error given by

2(2n+5)
— 1 )

. (19.35)

19.33 Tests of p or r based on the results given in the two preceding
sections take as the hypothesis that there is no correlation in the popula-
tion. For instance, suppose a value of t in a ranking of 15 was found
to be 0*6. For the standard erior we find, from (19.35), a value of 0- 19.
The observed value exceeds thrice this amount and is significant. Our
argument is as follows—

If there were no correlation m the population from which this ranking
is supposed to have been drawn as a sample, the order of appearance
of one variate is just as likely as any other order. Consequently, in
continued sampling we should, in the long run, obtain all possible rankings
of one variate with any particular ranking of the other. The population
of values of r so generated has a standard deviation given by (19.35).
Our observed value is very improbable m relation to this distribution, and
hence we suspect the hypothesis that the variates are independent.

19.34 But we have said nothing about the case when the variates are not
independent in the population and the foregoing results cannot be used
to test the difference of two rank correlation coefficients. Nothing appear^

456

THEORY OF STATISTICS

to be known on this point in relation to p, but some light has been thrown
on it in regard to r. In fact it may be shown —

(a) That the observed value of r is a good estimate of the value in the
parent population ;

{b)

That the standard error of r is not greater than

This limit is in some cases nearly reached so that no lower limit appears
possible* The test based on it may be rather insensitive but it seems
unlikely that any improvement can be effected unless some further
assumption is made about the nature of the parent population. (For
the further theory of this subject see Kendall's Rank Correlation Methods,
1948, Griffin).

SUMMARY

1. The following are the standard errors of the parameters named, the
parent population being assumed normal —

Variance

Standard deviation
Coefficient of variation
Correlation coefficient
Regression coefficient

V n

V2n

V I 2F2

l-r^

V 2 W ^ 10^

V n

c.Vl—r^ a
7 =^ or

1 2

cj^Vn

2. The standard error of the ^th moment measured about the mean is
given by

8. The correlation between errors in the qi\i and rth moments, both
measured about the mean, is given by

COV = T;

4* From the results of (2) and (3), and similar results for moments
about a fixed point, it is possible to calculate the standard error of any
function of the moments.

THE SAMPLING OF VARIABLES

457

5. In the normal population, errors in the mean and standard deviation
are uncorrelated.

6. In calculating the standard errors of moments the uncorrected
values should be used.

7. It IS unsafe to use the formulae for standard errors appropriate to the
normal population in cases where the population is suspected to differ from
the normal form , in particular, the formula foi the standard error of the

standard deviation,^— should not be used for parent populations which
are markedly lepto- or platy-kurtic.

8. Tests are given for the significance of the rank correlation coefficient
p and r when no parental correlation exists. When there is parent correla-
tion an upper limit to the standard error of r is given by

EXERCISES

19.1 In the weight distribution of Exercise 4.6, page 100, last column,
find the standard error of the standard deviation. Compare it with
the value obtained on the a^^sumption that the parent distribution is
normal.

19.2 In the same data, compare the ratio of the s.e. of the s.d. to the s.d.
with the ratio of the s.e. of the semi-interquartile range to the semi-inter-
quartile range.

19.3 Show that for a normal population the sfandard error of the s.d. is
less than the standard error of the semi-interquartile range.

19.4 In a sample of 1,000 the mean is found to be 17-5 and the standard
deviation 2*5. In another sample of 800 the mean is 18 and the standard
deviation 2-7. Assuming that the samples are independent, discuss
whether th^ tw^o samples can have come from populations which have
the same standard deviation.

19.5 Find the correlation between errors in the mean and standard devia-
tion for the height distribution of 8585 men of Table 4.7, page 82, and do
the same for the marriage distribution of Table 4.8, page 84.

19.6 Find the standard errors of the first four cumulants as calculated
from the moments.

19.7 Samples of 10,000 are taken from a normal population. For what
even moments does the standard error of the moment lie within 10 per
cent of the value of that moment ?

458

THEORY OF STATISTICS

19.8 For samples of (a) 100, (6) 1,000, draw a graph showing how the
standard error of the correlation coefficient from a normal population
varies with r.

19.9 (Data quoted by M. F. Hoadley, Note on the Association of
Relative Laterality of Hand and Eye from the Cambridge Anthropometric
Data,” Btometrika, 1928, 20B, 401.)

Three experiments were conducted to determine the relationship between
laterality of hand and laterality of eye. The correlations between (1)
difference of strength of grip and (2) difference in visual acuity were —

—0*02410 (3234 subjects)

—0 • 00738 (4003 subj ects)

+0*02962 (1447 subjects)

Find the standard errors of the three correlation coefficients, and hence
show that it cannot be concluded that there is any significant correlation
between laterality of hand and laterality of eye.

19.10 Find the standard errors of the partial correlation coefficients of
Example 12.1, page 290. Hence state whether any one is not significantly
different from zero, and if so, which. For the purpose of this exercise
normality may be assumed, although in all probability the actual data
do not emanate from a normal population.

CHAPTER TWENTY

THE DISTRIBUTION

20.1 In Chapters 17 to 19 we have seen that a knowledge of the sampling
distribution of a statistic gives us a means of judging from samples
the relationship between fact and theory. For instance, in Example 17.3,
page 389, we were able to infer from a knowledge of the binomial distribu-
tion that the dice which provided the data were probably biased ; and
in Example 18.6, page 428, we could apply a knowledge of the distribution
of the mean of samples from a normal population to reject the hypothesis
that the mean in the population was less than 67 inches.

In the present chapter we shall discuss a particular sampling distribution
of profound importance in statistical theory, and shall note its applications
to the testing of accordance between fact and hypothesis in a wide range
of cases.

Cells

20.2 In what follows we shall consider only data giving the frequencies
of individuals falling within various categories. Statistical data, as will
have been evident from the examples already given in this book, are very
often of this type.

Such data, whether relating to attributes or to continuous variates
or to a mixture of both, will in practice be arranged in compartments.
For example, in the association table on page 20 there are four com-
partments, corresponding to the four ultimate classes. In the table of
frequencies within various height ranges (Table 4.7, page 82), each range
determines a compartment, and the data consists of 8585 individuals
distributed in 21 groups.

It is convenient to have a name for these compartments. We shall
call them cells. The frequency falling in a cell will be referred to as the
cell frequency.

One and the same table may contain frequencies of more than one
order, and frequencies of different orders must be kept distinct. Thus
an association table has four cells with frequencies of the second order
and two sets of two (the border frequencies) of the first order. pxq
contingency table has pq cells of the second order (to condense our ter-
minology) and a set of p and a set of q of the first order. Each such set
must be considered by itself. The tests of this chapter are applicable

459

460

THEORY OF STATISTICS

to any homogeneous set, but not to a mixed '' set comprising cells of
different orders.

20.3 We shall denote the number of cells m the presentation of a set
of data by n, and the cell frequency occurring m the rth ceil by Thus,
in the table of page 82 we have, numbering the ceils downwards —

Wj = 2

Wg = 4
W3 ~ 14

W21 == 2

20.4 In the class of cases we shall consider, we wish to compare the
actual values m with the cell frequencies which would exist if a particular
hypothesis H were exactly verified. These latter values we shall denote
by the letter w, so that the theoretical frequency in the rth cell is m^,.

The cell frequencies m, are sometimes referred to as the “ expected
values on the hypotheses H. This is rather a special use of the word
expected/' in the sense we have already given, namely, that the w/s
assume the values which they would take if the hypothesis were exactly
verified for the particular set of data.

We shall wTite —

.... ( 20 . 1 )

so that the are the excesses of the actual over the expected frequencies.

Clearly the quantities x embody all the information in the data about
the discrepancies between theory and fact. If the a's are all zero, fact
and theory are in perfect agreement. If the ;r’s are large, the agreement
is poor.

Example 20.1. — As a simple example let us consider the 2x2 con-
tingency table of Example 2.5, page 25. Numbering the cells from left
to right we have —

— 276, Wg = 3

W3 = 473, ^4 = 66

Now let our hypothesis H be that inoculation and exemption from attack
are independent. If this be so, the expected frequencies are —

= 255*5, = 23*5

^ 493*5, == 45*5

and hence we have —

Xj = = 20*5, = —20*5

20*5, X4=^20*5

The x*s are, in fact, in this particular case, the numbers we referred to in
Chapter 2 as <^-numbers. We have already considered them as reflecting
the divergence of fact from theory.

THE DISTRIBUTION

461

Constraints

20.5 In the example we have ]ust considered, one important effect is to
be noted, viz. that when we have calculated one independent frequency,
say m^, the other three follow arithmetically from the fact that the two
frequencies in any row or column must add up to the border frequency
111 that row or column.

In fact, we have —

^1+^2 = Oj

“i~A^3 = 0 > . . . . (20.2)

^2+^4 =^o)

We need not add ^3 +^4=0, since this is given by the last two equations
in conjunction with the first. There are only three independent equations.

Thus, whatever our hypothesis H may be, the conditions of the problem
impose limitations, expressed by the equations (20.2), on the waj’’ in which
the m's and the :r’s may be chosen. If one ni or one is fixed by H, the
other three are determinate in accordance with the conditions of the
data themselves.

Similarly, suppose we wished to examine the height data of page 82
in the light of the hypothesis that the parent distribution, of which this
is a sample, is normal with given mean and standard deviation. With
the aid of the table of the probability integral we can determine the cell
frequencies on this hypothesis ; but again the problem imposes a limita-
tion on the way in which the theoretical cell frequencies are assigned,
namely, that they must add up to the total number 8585 of the sample.
When 20 frequencies are fixed, the other is determined by mere arithmetic.

20.6 In general, when the conditions of the problem impose limitations
of this kind on the number of cell frequencies which may be fixed by H
we say, borrov mg an expression from Statics, that they impose constraints.
In the example of the 2x2 contingency table there were three independent
constraints, expressed by the equations (20.2). In the case of the height
distribution there is one constraint expressed by the fact that the sum
of the cell frequencies must be 8585.

Linear constraints

20.7 Constraints which involve linear equations in the cell frequencies
(i.e. equations containing no squares or higher powers of the frequencies)
are called linear constraints. The two instances above are of this type.
Linear constraints are of paramount importance, and w^e shall shortly
confine our attention to them alone.

Degrees of freedom

20.8 We denote the number of independent constraints in a set of data
by /c. We then define the number v by the simple equation

V s= n— /C

462

THEORY OF STATISTICS

and call p the number of degrees of freedom of the aggregate of cells. It
IS the number of cell frequencies which can be assigned at will, the
remaining k following from the conditions to which the data are subject.

Thus, for the 2 x2 table /c=3 and for, as we have seen, the fixing
of one cell frequency fixes them all. For the height distribution /c=l,

j^=20.

Example 20.2. — Let us find the number of degrees of freedom of a
pxq contingency table.

The constraints of such a table are similar to those of the 2x2 table.
Thus the sum of the cell frequencies in each row is determined as being
the border frequency in that row, and similarly for the columns. Hence
each of the p columns and q rows imposes a constraint. From the total
p-\-q constraints we must, however, subtract one, for they are not
algebraically independent ; there is one relation between them, expressed
by the fact that the sum of the border column equals the sum of the
border row, namely, the total frequency N,

Hence there are p-\-q—\ independent linear constraints. Hence,

V = n—K

We might have got this result more directly by considering that the
cell frequencies in the first p—l columns and q-—\ rows are determinable
at will, the rest following automatically from the border frequencies.
Hence the number of degrees of freedom, being the number of cells which
can be so filled, is as before.

20.9 Now let us consider a set of data arranged in n cells, the total
frequency being iV.

The theoretical frequency in the rth cell is This means that the

chance of an individual failing into this cell is and the chance of its

not doing so is regard the actual frequencies m as

having been arrived at by distributing the N individuals among the
n cells in such a way that the chance of an individual falling into the
m

rth cell is Hence the probability that of the N individuals, fall
into the rih cell and the remainder elsewhere is the term involving

in the binomial

THE DISTRIBUTION

463

Thus, this binomial will give us the relative frequencies of the various
values which can take in different samples, of which the actual data
form one.

If N is fairly large and ^ is not small, this distribution is approxi-
mately normal with mean That is to say, is distributed normally
about a mean or is distributed normally about zero mean.

Definition of

20.10 We now define the quantity x^ ^7 equation

( 20 . 3 )

the summation being taken over the n cells.

The student can verify for himself that this definition is consistent
with that given in equation (3.4), page 52, for the particular case of
divergence from independence in a contingency table.

We can write ^ slightly different form. For

— 2S(Wy) -f S(w^)

-iV ^ . (20.4)

This corresponds to equation (3.7), page 53.

20.11 If are zero, and hence the actual cell frequencies

coincide with the expected cell frequencies. On the other hand, if some
or all ot the x's are large, x^ will be large.

It will thus be evident that x^ affords a measure of the correspondence
between fact and theory. It must not be forgotten, however, that it
ignores the signs of the x*s and hence takes no cognisance of certain
information which those signs may convey. We shall take up this point
again later.

20.12 If the use of x^ is to be satisfactory, we must be able to dis-
tinguish significant values from those which may have arisen by sampling
fluctuations. This leads us to inquire what is the probability of getting
a particular value of x^ from a set of chosen at random, and this in
turn leads to the question : What is the sampling distribution of ?

We shall not give a proof here of the important answer to this question,
but shall content ourselves with quoting it and indicating briefly the
method by which it is obtained.

464

THEORY OF STATISTICS

We have already seen that the sum of n normally distributed variates
is itself normally distributed (10.8). The sum of the squares of n normal
variates is not so distributed, however. In fact, the sum of the squares
of n normal variates, drawn from a population with unit standard devia-
tion and zero mean is distributed in a form given by the equation

£2

. . . (20 5)

where is the sum in question

Now it has already been shown that under the conditions assumed
the x's are each distributed normally about zero mean, and it may be
shown further that regarded as the sum of the squares of v

variates each distributed normally with unit s.d. and about a zero mean.
Hence the distribution of given by

y = .V ^ • • (20.6)*

20.13 It follows, as in 18.8, that if we take a random set of m’s and
calculate x^ from them, the probability of getting a value of x^ great
as, or greater than, this observed value Xo"> is the area of the curve (20.S)
to the right of the ordinate at Xo divided by the total area of the curve ,
or, in the language of the integral calculus,

(20.7) t

The curve, as we shall see later, extends from 0 to + 00 , which accounts
for the limits of the integral in the denominator of the above expression.

* Since the variate m this expression is x. "the distribution should, perhaps, be known
as the X"<tistribution, not the x"-distribution The latter name is, however, in universal
use, and the tables of the integral of equation (20 7) are usually prepared with argu-
ment X®

f The actual values of P are, expanding this integral,

, /2r“ /2

(f+rs

2d .

1.35

__ -2^:

■^135.

if V IS odd

( 1 .- 2 )

‘ + ' 2 + 0+2 4'6 + ■ •

■^ 2 . 4.6 . ( 1 .- 2 )

if V IS even

The first term of the first series may be obtained from the probability integral.
Values of P for given y? and v are provided in Tables for Statisticians and Biomeiricians,
a new^edition of which, in course of preparation, gives more detailed tables than have
hitherto been available.

THE DISTIUBI^TION

465

Tabulation of P for the distribution

20.14 The rather formidable result of equation (20.7) need occasion

no alarm to the student who is unacquainted with the notation and methods
of the integral calculus. The function P has been tabulated for certain
ranges of v and x^ same way as the probability for the normal

curve, and the tables are in most cases sufficient for the practical applica-
tion of the results of the present chapter. More convenient is the table
given in Appendix Table 3, which shows the values of x^ given values
of V and P.

20.15 It IS desirable to point out that other writers have used different
letters to denote the number of degrees of freedom. Karl Pearson, in
the tables to which we have just referred, used the number which is
one more than our v. R. A. Fisher writes n instead of our v, so that we
have —

V = n' — 1 (Pearson) = n (Fisher)

We have thought it desirable to introduce the symbol v in order to avoid
confusion with the use of n' and 71 as numbers in a sample or in a popula-
tion.

The x^ of significance when the theoretical cell frequencies are known

a priori

20.16 Armed with Appendix Table 3, we can now proceed as follows —
Having decided on the hypothesis to be tested, we calculate from it
the theoretical frequencies m^. (For the present we assume that this can
be done without reference to the observed frequencies The contrary
case will be considered later.)

From the and the w/s we calculate x^ according to (20.3) or (20.4).
We also ascertain v.

Then, from the table we determine whereabouts this value of in

relation to P.

The value P gives us the probability that on random sampling we should
get a value of x^ great as, or greater than, the value actually obtained.

Now, if P is small, our data give us an improbable value of x^* Thus
we have the alternative conclusions that either (^) an improbable event
has occurred, or {h) that the divergence of fact from theory is significant
of some real effect and cannot be attributed to fluctuations of sampling.
The smaller P is, the more we incline to the latter alternative ; if we do
decide to adopt it, the inferences we draw will depend on the nature of the
problem. Sometimes it will lead us to reject our hypothesis. Sometimes
it will lead us to suspect our sampling technique.

The following examples will illustrate the type of reasoning involved in
applying the ^^st.

466

THEORY OF STATISTICS

Example 20.3. — In some experiments on dice-throwing W. F. R. Weldon
rolled 12 dice 26,306 times, observing at eacb throw the number of dice
recording a 5 or a 6

If the dice are unbiased, the chance of getting a 5 or a 6 wdth one die
is Hence the chances with 12 dice of getting 12 5*s or 6*s, 11 5^s or 6's,
etc., are the successive terms in the binomial Hence the theo-

retical frequencies in 26,306 throws are the terms in 26,306
These are our

The following table shows the actual and the theoretical [m^
frequencies, together with the values of — —

ntf

TABLE 20.1'—12 dice thrown 26,306 times, a throw of 5 or 6 reckoned a success

Number of
successes

Observed

frequency.

{M)

Theoretical

frequency

(m)

flW — W

(w—

185

203

- 18

1'596

1,149

1,217

3-800

3,265

3,345

- 80

1-913

5,475

5,576

-101

1-829

6,114

6,273

-159

4-030

5,194

5,018

+ 176

6-173

3,067

- 2,927

+ 140

6-696

j 1,331

1,254

+ 77

4-728

1 403

392

+ 11

0-309

105

+ 18

3-724

10 and over

+ 4

1*143

Totals

26,306

35-941

Hence =35-941, and v=one less than the number of cells =10.

From the Tables for Statisticians and Biometricians we have, when
1^=10 (^'= 11 ),

P= 0*000857 for = 30
P= 0*000017 for x^==40

Evidently when x®=35*941, P will be extremely small. If we want to
evaluate it exactly we can proceed by the methods given in the Tables.
In fact P=0-000086.

Alternatively, from Appendix Table 3 we see that when %2--.23*209
and F=10, the value of P is 0*01. Thus P for %2=35*941 must be much
less than this value.

We may therefore say that the correspondence between theory and
fact is very poor. The extreme improbability of the observed event
enables us to say with some confidence that the divergence between the
two is significant, and hence that either our sampling technique or our
hypothesis is at fault. Now in this experiment Weldon took particular

THE DISTRIBUTION

467

care with the dice-throwing, and we may regard it as unlikely that there
was anything seriously wrong with the randomness of the sampling. We
are therefore led to doubt our hypothesis that the dice were unbiased.

Briefly, then, the test suggests that the dice were biased.

Example 20.4, — The following table shows the result of inoculation
against cholera on a certain tea estate —

TABLE 20.2

Not-attacked

Attacked

Total

Inoculated .

431

{427*7)

(S' 3)

436

Not-inoculated .

291

(294 3)

{5'7)

i 300

Total

722

i 1

736

We shall explain the figures in brackets presently. The question on which
we want to throw light is : Is there any significant association between
inoculation and attack ?

To answer this, let us take for our hypothesis H the supposition that
they are independent. If this is so, the expected frequencies, calculated
in the manner of Chapter 2, are those given in brackets. These we take
to be the the being the actual frequencies. We then have —

and

427-7^8*3

294-3

5 - 7 ^

= 3-27

V = 1

From Appendix Table 3 for 2*706, P=0-10 and for
P=:0-05. For our observed value of 3-27, P lies between 0-05and0-10.

Thus if H is true, our data give a result which would be obtained between
5 and 10 times in a hundred trials. This is infrequent, hut not very in-
frequent. Moreover, the theoretical frequencies in the '' attacked ”
column are not very large. We should therefore be unjustified in rejecting
H on this evidence, but we can say that the data lend some colour to the
supposition that H is not correct.

To sum up, the x^ f’^st shows that the data incline us, though not
strongly, to the belief that inoculation and attack are associated.

Example 20.5. — (Imaginary data.) An investigator into chocolate
consumption divided the United Kingdom into eight areas and took a

468

THEORY OF STATISTICS

random sample from each, the individuals so obtained being classified as
consumers or non-consumers of chocolate. His results were as follows —

TABLE 20.3

Area number

Total

Consumers .

(55)

(81)

142

(152)

(69)

(90)

(72)

100

(95)

142

(144)

758

Ncn-consumers .

(18)

(26)

(48)

(22)

(29)

(23)

(30)

(46)

242

Total

107

200

119

125

190

1,000

Do these results suggest that the consumption of chocolate varies
from place to place ^

Let us take as our hypothesis H the supposition that it does not, i.e,
that the two attributes in the above table are independent. The theo-
retical frequencies are then those shown in brackets, and we have —

12 02

similar terms

= 6-28

The table has* two rows and eight columns, and hence f~( 2 — 1 ) (8 — 1)~7.
From Appendix Table 3 we have for v— 7, P = 0*50; or

alternativelv, from the Tables for Statishcians and Biometricians for

v=.l (^^'= 8 ),

if =e, P =0-539750

if == 7 , p =0-428880

Hence, for ;^2=:0.28, P=0-51 approximately.

Thus there is no cause to suspect our hypothesis, and the data do not
suggest that the proportion of consumers of chocolate varies from place
to place, at least so far as this test is concerned.

Properties of the distribution
20.17 The curves

and the probability function P derived from them, have several interesting
properties which are worth noticing. As is essentially positive, we
consider only positive values of the variate.

(a) In the first place, it will be seen that when the curve is the
normal curve with unit standard deviation, for positive values of the
variate. Thus the test for v=l may be reduced to testing the significance
of deviations of a normally distributed variate.

THE DISTRIBUTION

469

(6) When V > 1 the curve is of the single-humped type. It is tangential
to the ;^-axis at the origin ^ maximum where x^~v—l and

then falls more slowly to zero as increases indefinitely. It is thus skew
to the right.

(c) As V increases, the curve becomes more and more symmetrical. In
fact, wh en v is large, \^2x^ is distnbuted approximately normally about a
mean V2i' — 1 with unit standard deviation. This result, due to R. A.
Fisher, enables us to dispense with tables of F for large values of v, say
V > 30, and to use the normal integral instead. In practice large values
of V are rather infrequent.

Example 20.6. — To find P when 04 ^=41.

We know that V23^ is distributed normally about mean \/82 — 1=9
with unit standard deviation. When x^=04, V2x^=l 1*314, which
therefore has a deviation 2*314 to the right of the mean. Hence we have
to find the area of the probability cuive to the right of the ordinate which
is 2*314 units to the right of the mean. From Appendix Table 2 this is
seen to be 0*0103 approximately.

Conditions for the application of the x^ ^^st

20.18 We may conveniently bring together at this point the various
precautions which should be observed in applying the distribution to a
test of significance.

[a] In the first place, N must be reasonably large. Otherwise the
are not normally distributed.

This IS a condition which is almost always fulfilled in practice. It is
difficult to say exactly what constitutes largeness, but as an arbitrary
figure we may say that N should be at least 50, however few the cells.

[h) No theoretical cell frequency should be small. Here again it is
hard to say what constitutes smallness, but 5 should be regarded as the
very minimum, and 10 is better.

In practice, data not infrequently contain cell frequencies below these
limits. As a rule the difficulty may be met by amalgamating such cells
into a single cell. Thus, in Example 20.3 above, the theoretical numbers
of throws with 10, 11 and 12 successes are (to the nearest integer) 13, 1
and 0. Instead of putting each into a separate cell we have run them
together into one cell 10 and over/'

(c) The constraints must be linear. The reason for this condition has
not emerged explicitly in the foregoing because we omitted the stage in
the proof of the x^ distribution at which it occurs.

20.19 To these three conditions we may add the following remarks,
which should also be borne in mind when the x® is being used.

{a) The x^ test tells us the probability of getting, on a random sample,
a value of x® equal to or higher than the actual value. If this probability

470

THEORY OF STATISTICS

IS small we are justified in suspecting a significant divergence between
theory and experiment.

We cannot proceed, however, in the reverse direction and say that if P
IS not small our hypothesis is proved correct. Ail that we can say is that
the test reveals no grounds for supposing the hypothesis incorrect ; or
alternatively, that so far as the test is concerned, data and hypothesis
are in agreement.

(6) Nor do only small values of P lead us to suspect our hypothesis or
our sampling technique. A value of P very near to unity may also
do so.

This rather surprising result arises in this way : a large value of P
normally corresponds to a small value of ^ dose

agreement between theory and fact. Now such agreements are rare —
almost as rare as great divergences.

We are just as unlikely to get very good correspondence between fact
and theory as we are to get very bad correspondence and, for precisely the
same reasons, we must suspect our sampling technique if we do. In short,
very close correspondence is too good to be true.

The student who feels some hesitation about this statement may like to
reassure himself with the following example. An investigator says that he
threw a die 600 times and got exactly 100 of each number from 1 to 6.
This is the theoretical expectation, and P=l, but should we believe
him ? We might, if we knew him very well, but we should probably
regard him as somewhat lucky, which is only another way of saying that
he has brought off a very improbable event.

20.20 At this point we can resume a topic which we laid on one side
in 20.11, namely the signs of the which are ignored by

It may happen that x^ quite a moderate value and P is not small
when ail the positive ^*s are on one side of the mode of the theoretical
distribution and all the negative x's on the other. There will thus be a
consistent “ shift " of the m's one way or the other from the m*s. This
may give us a value of the mean quite outside the limits of sampling.
Again, if the x*s are all negative in the cells farthest removed from the
mean, the standard deviation may show an almost impossible divergence
from expectation.

Thus, although the x^ f^st may reveal no cause to suspect the hypothesis,
a closer examination of the may.

Example 20.7, — Consider the following dice data (Table 20.4) (Weldon,
see Example 19.1.)

Now, in this example, all the x*s are negative up to 5 successes, positive
from 6 to 10 successes, and negative again for 11 to 12 successes. This is
almost one of the cases we referred to earlier in this section.

We have, in fact, already found (Example 17.8, page 389) that the
mean deviates from the expected value by 5* 13 times the standard error.

THE DISTRIBUTION

471

TABLE 20.4. — 12 dice thrown 4.096 times, a throw of 4, 5 or 6 points reckoned a

success

Number of
successes

Observed

frequency

{m)

Expected

frequency

(m)

4096(i4-i)“

w— w
(^)

{rh'-mY

- 1

1 0000

- 5

2-0833

- 6

0-5455

198

220

-22

2-2000

430

495

-65

8-5354

731

792

-61

4-6982

948

924

0-6234

847

^792

3'8194

536

495

i 41

3-3960

257

220

6-2227

0-3788

■;}.3

0-3077

Totals

4096

33 - 8104«;^2

From the tables we find —

V n' P

12 13 30 0-002792

12 13 40 0-000072

Hence, by simple interpolation for 33 . 3104 . 0-0018.

As a matter of fact, simple interpolation is of very little value for small values Oi'
P {cf. 24.12), and this value is wide of the mark, the true value being 0 • 00072 . Appendix
Table 3 shows us that P is less than 0-01.

From the extended tables of the normal integral in Tables for Statisticians
and Biometricians, Part /, we have —

Greater fraction of the area of a normal
curve for a deviation 5-13 . . , 0 9999998551

Area in the tail of the curve . . . 0-0000001449

Area in both tails .... 0-0000002898

so that the probability of getting such a deviation 1 + or — ) on random
sampling is only about 3 in 10,000,000.

Comparing this with the value of P, we see that the data are really more
divergent from theory than the ^^st would lead us to suppose.

20.21 Hence, if the signs of the x*s show any marked peculiarities,
it is as well to apply as many supplementary tests as are available, and
not to rely on the x^ ^^st alone. Such tests would include those for the
significance of the mean and standard deviation, which we have already
discussed.

Levels of significance

20.23 In the examples we have given above, our judgment whether P
was small enough to justify us in suspecting a significant difference between

472

THEORY OF STATISTICS

fact and theory has been more or less intuitive. Most people would agree,
in Example 20.3, that a probability of only 0-0001 is so small that the
evidence is very much in favour of the supposition that the dice were biased.
But we shall not always get such a decisive result. Suppose we had
obtained P=0* 1, so that the odds against the event are nine to one. Is
this value small enough to lead us to suspect the dice ? If it is not, would
P=0-01 be small enough ^ Where, if anywhere, can we draw the line ?

The odds against the observed event which influence a decision one
way or the other depend to some extent on the caution of the investigator.
Some people (not necessarily statisticians) would regard odds of ten to one
as sufficient . Others would be more conservative and reserve judgment
until the odds were much greater. It is a matter of personal taste.

20.23 There are, however, two values of P which are widely used to
provide a rough line of demarcation between acceptance and rejection of
the significance of observed deviations. These values are P=0*05 and
P=0-01, and are said to define 5 per cent and 1 percent levels of significance.
The value P =0-001, i.e. the 0 - 1 per cent level, is also used. A value of
P less than 0 - 05 will be said to fall below the 5 per cent level of significance,
and so on. The values of the 5 per cent and the 1 per cent levels, among
others, are tabulated in Appendix Table 3.

Example 20.8. — Let us consider the data of Exercise 2.11. In experi-
ments on the Spahlinger anti- tuberculosis vaccine the following results were
obtained. (As before, the figures in brackets are the independence values.)

Died or seriously

Unaffected or not

Total

affected

seriously affected

Inoculated , . .J

(8*87)

(10-13)

Not inoculated or mocu-j

8 i

lated with control media \

(5.13) ,

(5.87)

Total

Here,

^ 4.75 y

Fiom Appendix Table 3 we have when for P=0'05, x®=3-841, and
we have for P=0-01, %2== 6-635, so that P lies between the 5 per
cent level of significance and the 1 per cent level.

If, therefore, we take the 5 per cent level as appropriate to this case,
the results are significant ; but if we are more conservative and take the
1 per cent level, the results are not significant. In this particular case
the position is complicated by the relative smallness of the theoretical cell
frequencies.

THE DISTRIBUTION

473

The additive property of

20.24 It sometimes happens, by the repetition of experiments or other^
wise, that we have a number of tables for similar data from different
fields. The values of P for each may not be entirely conclusive. The
question then arises whether we cannot obtain a value of P for the aggre-
gate, telling us what is the probability of getting, by random sampling, a
series of divergences from theory as great as or greater than those observed.

The question is usually answered by pooling the results to form a single
table. But, apart from the fact that this is not always possible, we have
already seen (Chapter 3) that pooling is likely to introduce fallacies. A
better method is to proceed in accordance with the following general rule.

20.25 Suppose we have a number of groups of data, each furnishing a
X^ and a v. Add together all the s to form a single value and all
the v‘s to form a single value The x^ f^st may then be applied to Xi^
and as if they came from a single set of cells.

The validity of this rule will be evident when we consider how the x^
test was arrived at. The variate jJt in every cell is normally distributed
about a mean m, and Xi^ the sum of the squares of quantities like

— just as x^ was. .This, together with the linearity of the constraints,

which remains, was the essential part of the proof of the x^ distribution,
and hence the test remains true for Xi^

Example 20.9. — In Example 20.4 (inoculation against cholera on a
certain tea estate) we saw that the x^ although suggesting that
inoculation had some effect in immunising, did not allow us to place any
great confidence in such a conclusion. The following data give x^ ^tncl P
for six estates, including the one we have already discussed —

9-34 0-0022

6*08 0*014

2*51 0*11

3*27 0*071

5*61 0*018

1*59 0*21

Total 28*40

Here only one value of P is less than 0*01, and we might be inclined to
doubt whether the association between inoculation and immunity is reah
Let us, however, add the values of of v. We get Xi^~28-40 and

^ 1 = 6 , there being one degree of freedom from each of the six tables.

From Appendix Table 3 we see that this value is well beyond the one
per cent, significance point. If we require greater accuracy, from the
tables wc have —

474

THEORY OF STATISTICS

28 0*000094

29 0*000061

Whence by interpolation P =0*00008 approximately, i.e. we should expect
to get a great as this only 80 times in a million. We can, therefore,
regard the results, taken together, as significant with a high degree of
confidence.

Estimation of theoretical frequencies from the data

20.26 Our theoretical frequencies m may be calculated partly on the
basis of information from the data, partly on a priori grounds. Thus,
in the dice-throwing data of Example 20.3, our hypothesis that the dice
were unbiased enabled us to say that the chance of getting a 5 or a 6 was
I, and hence that the chances with 12 dice were the terms in 26,306 (f +
Here we take only the value of N, the total frequency, from the data.

In the association and contingency tables, the values of row and
column totals, as well as N, are taken from the data and we assume
a priori that the attributes are independent.

It may be, however, that we draw further information from the data
themselves in fixing the theoretical frequencies. In such cases an im-
portant modification is necessary in the previous methods of work, for the
number of degrees of freedom is further restricted by each piece of
information drawn from the data, as we have already seen for contingency
tables.

20.27 Consider, for example, the dice-throwing data of Example 20.3.
We have already seen that the dice were probably biased, so that the
chance of a success was not What, then, was it ?

To answer this question w^e can only appeal to the data. The propor-
tion of 5's and 6's in the total number of throws of individual dice
(26,306 X 12) was 0*3377. Let us therefore take this to be an estimate of
the true probability. We can be confident that it will be somewhere
very close, omng to the large number in the sample. The theoretical
frequencies will then be the terms in 26,306 (0*6623+0*3377)^2,

To take a second case ; consider the height distribution of Table 4.7
page 82. We have already had reason to suspect that this is a sample
from a normal population. we suppose this hypothesis* to be correct,
the question arises. What is the mean and standard deviation of the
population ? Here again we must estimate these quantities from the data,
in the manner of Chapter 18.

20.28 We shall denote values of the theoretical frequencies which are
calculated from parameters estimated from the data by the letter m', and
the value of calculated from them by so that we have —

THE DISTRIBUTION

475

Now, estimate of x^ if fho m''s are close to the w's, x'^ will

be close to x^- is made up of two parts, one measuring the divergence
betw^een theory and fact, the other due to errors of estimation of x^*
the second is small compared with the first, we may expect that the
test, applied with x'^ instead of the unknown x^> will continue to reveal
significant differences between theory and fact where such exist.

20.29 The question as to the precise conditions under which the
test is applicable for such cases has not been completely answered, but
it has been shown that, if the cell frequencies are large, the test still
applies subject to the following conditions —

{a) The number of degrees of freedom must be reduced by unity for
each constant of the population which is estimated from the data.

(6) The estimates must be of the type known as efficient.’'

We shall not be able in this Introduction to go into the theory of this
important class of estimate, but it will be sufficient if we indicate that the
estimates of the mean of a normal population, and the parameter m of the
Poisson distribution, are “ efficient " if calculated in the ordinary way,
i.e. by taking the value of the parameter in the sample to be the value of
the parameter in the population.

Example 20.10. — Reverting to the data of Example 20.3, let us estimate
the true chance of getting a 5 or a 6 from the data themselves. The
frequency of the successful event is 0*3377 of the whole. This is an
efficient " estimate of the chance. The following table gives the
observed frequencies and the theoretical frequencies calculated from the
formula 26,306 (0 *6623+0 •3377)i2_

TABLE 20.5. — 12 dice thrown 26,306 times, a throw of 5 or 6 reckoned a success

Number of
successes

Observed

frequency

(m)

Theoretical

frequency

{m')

m—m*

(m — m^y
m*

185

187

- 2

0-021

1.149

1,146

0-008

3,265

3,215

0-778

5,475

5,465

0-018

6,114

6,269

-155

3-832

5,194

5,115

1 -220

3,067

3,043

0-189

1,331

1,330

0*001

403

424

- 21

1*040

105

0-844

10 and over

0-250

Total

26,306

8*201

476

THEORY OF STATISTICS

Thus There are 11 cells, with one linear constraint. We

have also fitted one constant from the data, and hence we must take
v=9.

From Appendix Table 3 we then see that P is very close to 0 • 50. Thus
our hypothesis is now, so far as the concerned, in agreement

with experiment.

Experiments on the x^ distribution

20.30 Several statisticians have conducted experiments to verify the
theory which we have discussed in the foregoing sections. A certain
amount of work in this field remains to be done, but generally it may
be said that experiment supports the theory. So far as cases where the
m*s are calculated a priori are concerned there is little doubt of its
correctness.

In one set of experiments (by Yule) 200 beans were thrown into a
revolving circular tray with 16 equal radial compartments and the number
of beans falling into each compartment was counted. I he 16 frequencies
so obtained were arranged (1) in a 4x4 table, and (2) in a 2x8
table, x^ was calculated from the independence frequencies, as in
Example 20.5.

The experiment and the calculations were repeated 100 times. The
following table exhibits the actual and the theoretical distribution of x ^ —

TABLE 20.6. — Theoretical distribution of calculated from independence values, in
tables with 16 compartments, compared with the actual distributions given by 100

experimental tables

In the first case v must be taken as 9, in the second as 7

4 Rows, 4 Columns

2 Rows, 8 Columns

Expectation

Observation

Expectation

Observation

0- 5

16-6

34-0

29*5

5-10

48-4

47-1

56-5

10-15

26*0

15-3

15-20

7-3

3-0

20-

1-8

0-6

Total

100-1

100

100-0

100

In a second experiment with 2x2 tables 350 experimental tables of
100 observations each were available. Table 20.7 shows the actual and
theoretical distributions in this case.

IHE DISTRIBUTION

477

TABLE 20.7. — Theoretical distribution of for a table with 2 Rows and 2 Columns,
when calculated from the independence values, compared with the actual results
for 350 experimental tables

Value of x“

Number of tables

Expected

Observed

0 -0-25

134 02

122

0 25-0 50

48-15

0-50-0 -75

32-56

0 75-1-00

24-21

1 -2

56 00

2 -3

25 91

3 -4

13-22

4 -5

7-05

5 -6

3-86

5 01

Total

349 99

350

It IS interesting to see what happens if we apply the test to these
tables.

In Table 20.6, grouping together the frequencies from upwards,

so that IS found to be 2*27 for the 4x4 tables and 4*36 for the

2x8 tables, giving P=0-52 in the first case and 0*22 in the second.

In Table 20.7, P==0-58.

Goodness of fit

20.31 The x^ distribution, as we have seen, leads to tests of the corre-
spondence between theory and fact, and this and other reasons have
led to its being described as a test of the goodness of fit.*' This expres-
sion may be used in two ways. In the first place, it may describe the
fit '' of observed and hypothetical data. In the second, it may be used
without reference to a hypothesis merely to provide an objective method
of estimating the merits of a particular formula or a particular curve in
graduating a set of values or a series of points.

The arithmetic in the second class of cases is exactly the same as in
the first. Conventionally, we regard very low values of P as denoting
a poor fit, and moderate values as denoting a reasonably good fit. High
values show an excellent fit, and in considering them we take no heed of
the point discussed in 20.19 (6), since we are assessing the closeness of
the curve to the data, not the probability that the first represents a popula-
tion from which the second was derived by random sanapling.

478

THEORY OF STATISTICS

SUMMARY

m )

~-N

where m refers to the observed and m to the theoretical frequencies.

2. The number of degrees of freedom of an aggregate of cells is denoted
by V, and is equal to the number of cells whose frequencies can be deter-
mined at will. When v cell frequencies are determined, the remainder are
calculable directly from the conditions to which the cell frequencies are
subjected by the nature of the data.

3. The frequency-distribution of is given by

4. From this it is possible to ascertain the probability P that on
random sampling we should get a value of ^.s great as or greater than
a given value. Tables have been constructed for this purpose.

5. The x^ distribution may be applied to data grouped in cells provided
(a) that the total number N in the sample is large, (b) that no theoretical
cell frequency is small, and (c) that the constraints are linear.

6. The value of P for any given case enables us to judge of the corre-
spondence between hypothesis and data.

7. When the theoretical cell frequencies have to be calculated from
parameters estimated from the data, the x^ test can be applied with

y'2

^ m'

instead of x^* provided that the cell frequencies are large, the estimates
are '' efficient,*' and the number of degrees of freedom used in ascertaining
P is reduced by unity foi every parameter which is estimated.

8. The value of P can also be used to give an objective criterion of the
** goodness of fit " of a curve .to a set of points or of a formula to a set of
values.

EXERCISES

20.1 The following table (Weldon) gives the results of a dice-throwing
experiment : —

12 dice thrown 4,096 times, a throw of 6 reckoned a success

Number of successes .0 1 2 3 4 5 6 7 and over Total

Frecjuency . . 447 1145 1181 796 380 115 24 8 4096

THE DISTRIBUTION

479

Find hypothesis that the dice were unbiased and hence show

that the data are consistent with this hypothesis so far as the ^^st
is concerned.

20,2 Perform an experiment by throwing a die 600 times and noting the
number of points at each throw. Use these data to inquire whether the
die is biased.

20.S 200 digits were chosen at random from a set of tables. The fre-
quencies of the digits were —

Digit . . ,0123456789 Total

Frequency . . 18 19 23 21 16 25 2 2 20 21 15 200

Use the f^^st to assess the correctness of the hypothesis that the digits
were distributed in equal numbers in the tables from which these were
chosen.

20.4 Perform an experiment on the lines of Exercise 20.3 by taking, say,
the last figure in 200 logarithms taken from a set of five-figure logarithm
tables,

20.5 (Data : Yule, Jour, Anthrop. InsL 1906, 36, 325) Sixteen pieces
of photographic paper were printed down to different depths of colour
from nearly white to a very deep blackish brown. Small scraps were
cut from each sheet and pasted on cards, two scraps on each card one above
the other, combining scraps from the several sheets in all possible ways,
so that there were 256 cards in the pack. Twenty observers then went
through the pack independently, each one naming each tint either “ light,”

medium '' or '' dark.”

The following table shows the name assigned to each of the two pieces
of paper —

Name assigned to
Lower tint

Name assigned to upper tint

Light Medium Dark

Total

Light .

850

571

580

2001

Medium

618

593

455

1666

Dark .

540

456

457

1453

Total

2008

1620

1492

5120

Show that there is a significant association between the name assigned
to one piece and the name assigned to the other.

20.6 Apply the x^ "the data of Example 2.8, page 29, and examine

the justification for the conclusions there drawn.

480

THEORY OF STATISTICS

20.7 Show that, if v is large, P is below the 5 per cent level of significance

V^2-v'2f-1>1-65

and below the 1 per cent level of significance if

V^-V2f-1>2-33

20.8 Table 3.6, page 64, gives the number of criminals of normal and.
weak intellect for various ranges of weight.

Assuming this to be a random sample of criminals, do the data support
the suggestion that weak-minded criminals are not underweight ?

20.9
c d‘

Show that in a 2x2 contingency table wherein the frequencies are
calculated from the “ independence ” frequencies is

-\-d) (a -)-c)

20.10 Show similarly that for a 2 xm table

where are the 2 frequencies in the rth column and iVi, iVg are the

marginal sums of the 2 rows.

20.11 Two investigators draw samples from the same town in order to
estimate the number of persons falling in the income groups “ poorer/'
middle class," " well to do." (The limits of the groups are defined in
terms of money and are the same for both investigators.) Their results
are as follows —

Investigator

“ Poorei ’

Income group

Middle Class" " Well to do "

Totals

140

100 15

255

140

50 20

210

Totals

280

150 35

465

Show that the sampling technique of at least one of the investigators is
suspect.

THE DISTRIBUTION

481

20.12 Exercise 8.17 gives the number of deaths per day of women over
85 published m The Times during 1910-12. Using the theoretrical
frequencies obtained in that exercise on the hypothesis that the numbers
are distributed in a Poisson series, employ the x^ to estimate the
correctness of this hypothesis.

20.13 Design and execute an experiment involving the x^ ^^st to test
the randomness of a set of random sampling numbers.

20.14 (Data: G. Mendel's classical paper on Experiments in Pi ant-
Hybndisation ” — quoted in translation in W. Bateson's MendeTs
Principles of Heredity T)

In experiments on pea-breeding, Mendel obtained the following fre-
quencies of seeds : 315 round and yellow , 101 wrinkled and yellow ;
108 round and green ; 32 wrinkled and green. Total 556.

Theory predicts that the frequencies should be in the proportions
9 : 3 : 3 : 1.

Examine the correspondence between theory and experiment.

20.15 A particular experiment gives, on hypothesis H, v=8;

when repeated it gives the same result. Show that the two results taken
together do not give the same confidence in H as either taken separately.

20.16 (Data from the Registrar-General's Statistical Review for England
and Wales, 1941, Tables, Part II, Civil), The following figures show the
number of births in England and Wales in 1941 by month of occurrence —

January

50,159

July

49,395

February

45,885

August

50,443

March

50,819

September

51,562

April

49,070

October

50,224

May

50,771

November

47,168

June

46,788

December

50,529

Total 592,813

Use the x^ f^^st to discuss whether there is any seasonality in birth revealed
by these data.

CHAPTER TWENTY-ONE

THE SAMPLING OF VARIABLES

SMALL SAMPLES

The problem

21.1 We now proceed to examine the theory of samples which are not
large enough to warrant the assumptions underlying the work of Chapters
17 to 19. In particular, it will no longer be open to us to assume {a)
that the random sampling distribution of a statistic is approximately
normal, or even unimodal, or (b) that values given by the data are
sufficiently close to the population values for us to be able to use them in
gauging the precision of our estimates.

The removal of these assumptions imposes severe restriction on our
work, and, as we shall see, an entirely new technique is necessary to deal
with the problems for which they are not permissible. The division
between the theories of large and small samples is therefore a very real
one, though it is not always easy to draw a precise line of demarcation.
We should point out, however, that as a rule the methods of the theory
of small samples are applicable to large samples, though the reverse is
not true.

Estimates

21.2 In the theory of large samples we were able to take as an estimate
of a parameter in a population the value calculated from the sample as if
it were itself the population. This procedure, obvious though it seems,
IS not in general valid for small samples. We must therefore discuss
briefly the basis on which estimates of given parameters are to be made.

A full investigation of this question would take us far beyond the limits
of this book. It involves matters of considerable mathematical and
philosophical complexity, some of which still form the subject of dispute
among statisticians. But in the theory of small samples tlie main para-
meters of interest are the mean and the standard deviation (or the
variance), and we will proceed to consider these two.

Estimates of the arithmetic mean

21.3 We shall take as the estimate of the arithmetic mean the value
of the sample mean. That is to say, if we have n sample values

X 2 > • * • our estimate x of the mean in the population is

X = hi{x)

n ^ ^

482

. ( 21 . 1 )

THE SAMPLING OF VARIABLES 483

For estimates of the mean, therefore, the practice is the same for small
samples as for large.

It may be shown that for samples from a normal population an estimate
obtained in this way is the best '' m the sense that its sampling variance
is less than that of any other estimate of the mean.

Estimates of the variance

21.4 Let us denote the variance in the population by and the mean
by m.

If m is known, we take as an estimate of the variance the mean square
deviation of the sample about m ; i.e. the estimate, which we write as s®,
is given by

= .... ( 21 . 2 )

In general, however, we do not know the value of m, which will itself
have to be estimated. In this case equation (21.2) is no longer applicable.

21.5 If m is the population mean and x is the sample mean, we have —

=^T,{x--x+x—m)^

= l!i{x--x)'^-^n[x--‘my
Hence, -

= !^{x-xY-\-{x-mY

The term '^{x—xY is the variance of the sample. We see that

it differs from by the term [x—my.

Now this term will not, in general, vanish ; nor will it vanish on the
average in a large number of cases, for it is essentially positive. Hence,
if we take the variance of the sample to be an estimate of the variance
of the population we shall involve ourselves in a systematic error of magni-
tude (^— m)^.

This term is the square of the deviation of the mean of the sample
from the mean of the population, and its average value in a large number of
samples is the variance of the mean, which we know to be equal to
It seems reasonable, therefore, instead of ignoring the presence of the
term to take it as equal to cy^jn. We will attempt, on this basis,

a new estimate, which we shall write s' We have then —

n ^ ' n

484

THEORY OF STATISTICS

The value of cj is unknown, but we may, as an approximation, write
instead. If we do so we get —

n ^ ^ n

.... (21.3)

The effect of taking $'^ given by equation (21.3), instead of the vaiiance
of the sample, will thus be to eliminate the systematic error of estimation
to which we have just referred.

21.6 We may look at this in a slightly different way. Suppose we
take a large number of estimates of the variance of a population compiled
according to equation (21.2), m being assumed knowm. These estimates
will fall into a distribution which is the sampling distribution of the
variance in samples of n. If, as will usually be the case, it is of the uni-
modal type, we expect it to have a mean located at the true value of
the variance in the population.

Now if we take as estimates of the variance the variance of the samples
(each about its own sample mean), the above will not be true, owing to
the small systematic shift represented by the term {x—m )^ ; but it will
be true of the estimates given by equation (21 3), and this is therefore
a preferable estimate to take.

21.7 Equation (21.3) was obtained by reasoning which does not depend
on the size of and strictly speaking we should take it as applicable
also to large samples But if n is large, n and n — 1 are for all practical
purposes equal. With such samples our results are true only within the

range of the standard error, which is usually of order and there is

little point in straining after an illusory refinement by taking 1 instead
of n in calculating the variance.

From a similar point of view it might be thought that since the term
0 ^ jn is generally less than the square of the standard error of the variance,
it is equally idle to make allowance for it in estimating the variance.
This would be true if the term were zero on the average ; but in fact it
is not, being a biased error, and w^e are justified in the long run in allowing
for it.

Furthermore, we may point out that the use of the corrected
value obtained by allowing for the term jn, is jDnly valid on the average.
If, on random sampling, we get a sample variance greater than the popula-
tion variance, the correction only makes matters worse, and may even
lead to an absurd result.

Degrees of freedom of an estimate

21.8 In discussing the ^®st we introduced the notion of number of

THE SAMPLING OF VARIABLES

485

degrees of freedom, being the number of cells in an aggregate whose fre-
quency could be assigned at will. We may conveniently extend this
nomenclature to estimates of parameters and particularly of variance.

We shall refer to the divisor in the estimates of equations (21.1), (21.2)
and (21.3) as the number of degrees of freedom of the estimates, and
shall write it as v. Thus, v in equation (21.2) is n, and in equation (21.3)
is n — l.

That this convention conforms to that adopted for the test may
easily be seen. We saw that v is the number of cells, that is, the number
of terms contributing to the sum, less one for each constraint and one
for each parameter which had been estimated from the data. In the
quantity there are n independent contnbutions of the type

{x—mY, and hence we may say that n is the number of degrees of freedom
of that estimate ; but m the quantity we have used the data to

estimate x, and hence the number of degrees of freedom is lowered by
unity, i.e. equals n — l.

Test of significance

21.9 It cannot be over-emphasised that estimates from small samples
are of little value in indicating the true value of the parameter which is
estimated. Some estimates will be better than others, but no estimate is
very reliable. In the present state of our knowledge this is particularly
true of samples from populations which are suspected not to be normal.

Nevertheless, circumstances sometimes drive us to base inferences,
however tentatively, on scanty data. In such cases we can rarely, if ever,
make any confident attempt at locating the value of a parameter within
serviceably narrow limits. For this reason we are usually concerned, in
the theo^ of small samples, not with estimating the actual value of a
parameter, but in ascertaining whether observed values can have arisen
by sampling fluctuations from some value given in advance. For example,
if a sample of ten gives a coirelation coefficient of +0*1, we shall inquire,
not the value of the correlation in the parent population, but, more
generally, whether this value can have arisen from an uncorrelated
population, i.e. whether it is significant of correlation in the parent.

21.10 The remainder of this chapter will accordingly be devoted to a
brief discussion of various tests of significance. Within this book we
shall not have space to deal with these tests as fully as we should like ; but
our account of sampling methods would be incomplete without some
reference to sundry results of great intrinsic interest and importance in
the field of small samples.

The assumption of normality

21.11 We have already considered one test of significance, that given

by the distribution of x^- is one of the simplest and most general

tests known ; but the student will recall that it depends on the assumption

486

THEORY OF STATISTICS

that the theoretical distribution of cell frequencies in each cell is normal.
This is justified under the conditions laid down in 20.18.

In the tests which we shall now discuss we are similarly compelled to
make some assumption about the nature of the parent population, although
we shall no longer be able to lay down analogous conditions on the arrange-
ment of the data under which the assumption is justified. We shall
specifically assume that the parent population is normal unless otherwise
stated.

21.12 Our results will, therefore, be strictly true only for the normal
population. Some experiments have been made to throw light on the
question whether they are true for other types of population. It appears
that, provided the divergence of the parent from normality is not too
great, the results which are given below as true for normal populations are
true to a large extent for other populations. Theoretical work confirms
that the results remain true for populations which do not deviate
markedly from normality ; but if there is any good reason to' suspect that
the parent is markedly skew, e.g. U- or J-shaped, the methods of the
succeeding sections cannot be applied with much confidence.

21.13 We may direct attention to one further point on which caution
is necessary. In the theory of large samples we recommended the student
to base his conclusions on a range of six times the standard error, and
pointed out that for normal populations the probability of deviations from
the true value outside this range was less than 3 in 1,000. One can feel
great confidence in conclusions supported by probabilities of this order.
But in the theory of small samples it is, as a rule, necessary to use larger
probabilities, say, of one in 20 or one in 100, e.g. the 1 per cent and 5 per
cent levels of P in the ^®st. The force of inferences based on prob-
abilities of this order is not so great as before, and the student should bear
this fact in mind.

21.14 For a known parent population, and in particular for a normal

parent, it is not difficult to find expressions for the random sampling
distribution of the commoner statistics such as the mean and standard
deviation. But these distributions, even when mathematically tractable,
will in general contain certain parent values. For instance, the sampling
distribution of the means of samples of n from a normal population with
mean m and standard deviation a is also normal with mean m and standard
deviation In the cases which we wish to consider, n is not large

enough for us tp take estimates of m and a from the sample to find the
sampling distribution to any close degree of approximation.

It is, however, a remarkable fact that we can construct certain statistics
whose sampling distributions are either independent of, or dependent
on only one of, the constants of the parent. We will proceed to consider
two important distributions of this kind, the so-called ^distribution, due
to Student," and the 5 ;-distribution, due to R. A, Fisher.

THE SAMPLING OF VARIABLES

The ^distribution

21.15 Writing, as before.

487

^ = h{x)

n ^ ^

s'* =

n— 1 ^ '

iet us define a new statistic t by the equation

<=^Vh^ .... (21.4)

where p=w~-l and m is the mean of the population.

We shall refer to v as the number of degrees of freedom of t.

Then it may be shown that, for samples of n from a normal population,
the distribution of t is given by

= ■ ( 21 . 5 )

(‘+t) ’

21.16 We will imagine chosen so that the area of the curve given
by equation (21.5) is unity. Then, precisely as for the distribution,
the probability that, on random sampling, we shall get a value of t not
greater than some value is the area of the curve to the left of the ordinate
at the point We may write this

( 21 . 6 )

Similarly, the probability that we get a value of t between the limits
and /g is given by

y^dt

1 +

y+i

1 )'

( 21 , 7 )

Form of “ Student’s ” distribution

21.17 The curves given by equation (21.5) are easy to siady. Clearly
they are symmetrical about t=0, since only even powers of t appear in their

equation.

Further, since

decreases as t increases, the curves

will have a mode (coinciding, of course, with the mean) at if =0, and will
tail off to infinity on each side. They will, in fact, be symmetrical single-
humped curves rather like the normd curve, only more leptokurtic.

488

THEORY OF STATISTICS

As V tends to infinity,

V + 1
2

- tends to e and hence t is dis-

tributed normally. This fact enables us to use the tables of the normal
integral to evaluate P approximately when v is large.

21,18 At the end of this book we reproduce by permission tables of
the integral (21.6) calculated by Student himself (Appendix Table 4).
These have been reduced to three places of decimals from the original four.

Tables of rather a different form have been given in the Fisher-Yates
StaUstical Tables and in Tables for StaHsHctans and Btomeirictans, Part /,
and to avoid possible confusion we point out where these tables differ.
Tables for Statisticians, etc., gives the values of

_ po y^z

V + 1

where for v from 1 to 9. These values (which were also calcu-

lated by Student are of the same kind as, but more limited in range
than, those of our table.

The Fisher-Yates tables adopt the standpoint we have already noticed
m discussing the yf distribution (Chapter 20), and gives values of t
corresponding to various values of v and the 5 per cent and 1 per cent
levels of a third probability P^.

Ps and Pj. are simply related. P^ is the probability that an observed
value will not exceed P^ is the probability that an observed value of t,
regardless of sign, will exceed

Hence,

Ps = Area of curve to the left of ordinate Iq
P jp == Area to right of t^ -f area to left of —Iq

= 2 (Area to right of Q (since the curve is symmetrical)

= 2(1-P,) ......... (21.8)

The student should keep these relations in mind, particularly when
thinking of levels of significance. In the sense of Fisher and Yates, a
value of Pp will fall below the 5 per cent level if Pp is less than O-OS.
This implies that Pq is greater than 0*975, not 0*95.^

^ A compansoa of the tables is not made any easier by the fact that “ Student "
and Fisher use n to denote the degrees of freedom, whereas Tables fov Statisticians uses
it to denote the number in the sample It is probable that future editions of Tables
for Statisticians will give more complete tables for the percentage points of t

The distinction between Pg and Pf did not arise in Chapter 20 because x® is essentially
positive.

THE SAMPLING OF VARIABLES

489

Applications of “ Student’s ” distribution

21.19 We proceed to give one or two examples of the way in which
the Student distribution is generally used to test the significance of
various results obtained from small samples.

Example 21.1. — ^Ten individuals are chosen at random from a popula-
tion and their heights are found to be, in inches, 63, 63, 66, 67, 68, 69,
70, 70, 71 and 71. In the light of these data, discuss the suggestion
that the mean height in the population is 66 inches.

In the first place, let us note that the population is likely to be approxi-
mately normal, from our knowledge of height distributions, and the
sampling is random.

In the sample we find that

and

X =67-8 inches
s' = 3*011 inches

Let us now calculate t from equation (21.4), taking m to be 66 inches.
We have —

t =

67*8-66

3*011

Vio

1*89

From the Appendix Table 4 (column v = 9) —

Hence,

for ^ = 1*8,
for ^ = 1*9,

for f = 1*89,

P = 0*947
P =0*955

P = 0*954

Thus the chance of getting a value of t greater than that observed is
1 —0*954, i.e. 0*046, or about one in twenty. The probability of getting t
greater in absolute value is 0*092, or about one in ten. We should hardly
regard this as significant ; but if we did, we should argue that as the
observed value of t is improbable, the initial assumptions on which we
obtained it were incorrect; and this in turn suggests that there is some
doubt about the true mean being 66 inches.

Example 21.2. — (Voelcker's data quoted by '' Student," Biometrika,
1908, 6, 19.)

Voelcker grew certain crops of potatoes dressed (a) with sulphate of
pbtash, and (b) with kainite. In four experiments, two of each of 1904
and 1905, the differences in yields per acre (sulphate plot less kainite
plot) were—

0*5464 ton
0*3013 „

1*5241 „

0*6786 „

490

THEORY OF STATISTICS

This suggests that sulphate of potash is a better manure than kainite.
Required to discuss the question.

From our knowledge of crop yields we expect them to be distributed
in a unimodal form not very far removed from the normal. Let us
suppose that the two manures have the same effect on yield. Then the
differences of plots will be distributed in an approximately normal form
about zero mean.

The mean of the four differences is 0*7626 ton, and we find s' =0*5312.

Hence,

0*7626 -0
0*5312
2*871

From the tables, for v=3, P =0*968 approximately.

Hence the chance P of getting a value of t greater than that observed
is about 1 in 33. The chance of getting a value greater absolutely than
the observed value is 0*06. If we choose to regard this as significant,
we are led to suspect our hypothesis that the two manures exert equal
influences on yield, and hence to suppose, though with little confidence
so far as these data are concerned, that sulphate of potash is the better
manure.

21.20 The student who wishes to apply the ^distribution for himself
is advised to make a careful study of the logic of the argument under-
ling the inferences we have drawn in the foregoing two examples.

In Example 21.1 we saw that the chance of getting a value of t less
than 1 *89 is approximately 0*954. This is not the same thing as sa^ng
that the probability of a deviation in the sample mean of 1 *8 inches or
less is 0.954. In fact, we do not know this probability, and the smallness
of the sample prevents us from approximating to it with any closeness.
It might happen that a in the population was such that a deviation of
1*8 inches was not at all improbable. The relative improbability of t
would then be due to deviations of s' from a.

Comparison of two samples

21.21 Suppose we have two samples ^^2 . * . and . . . x\^.

Let us. as before, define

. (21.9)

THE SAMPLING OF VARIABLES

491

Let us further define

. . ( 21 . 10 )

If the two samples come from the same population, 5 '^ will be an estimate
of a^. It has, as we might expect, degrees of freedom, since

both % are calculated from the data.

Let us write

2 .... ( 21 . 11 )

and define

, (21.12)

s ' V %+^2

Then it may be shown that t, as so defined, is distributed according to
the form of equation (21.5) with v degrees of freedom.

Example 21.3. — (Data from R.A. Fisher, Matron, 1925, 5, 95.)

Eight pots growing three barley plants each were exposed to a high
tension discharge, while nine similar pots were enclosed in an earthed wire
cage. The numbers of tillers in each pot were as follows —

Caged . . ,17, 27, 18, 25, 27, 29, 27, 23, 17

Electrified . • 16, 16, 20, 16, 20, 17, 15, 21

We are interested in the question whether electrification exercises any
real effect on the tillering.

We find

= 23-333 = 17-625

,'2

x^—x^ =5*708
221-875 = 14*7916

^^5-708 /i

'“"3-846V'

^ 8x9

s' =3*846
3*05

V = 8+9-2 = 15

From the tables we find that = 0*996.

Hence, if the samples came from the same population they furnish a
value of t which is improbable — an absolutely greater value would arise
only 8 times in a thousand. We therefore suspect that the populations
are different, i.e. that electrification does exert some efiect on the tiULering.

492

THEORY OF STATISTICS

21.22 In applying the ^distribution to two samples as in the preceding
example one further point should be borne in mind. It does not follow
from a significant value of t that the samples come from populations which
have different means. Samples from two populations with the same
means and different standard deviations would also furnish significant
Vs on occasion. We can test whether this is so by the method of 21.27
below.

Significance of regression coefficients

21.23 From (21.4) it is clear that “Student's” t is a. ratio, being, apart
from constants, the ratio of the estimate of the sample mean (measured
from the parent mean) to the estimated standard deviation. The
simplicity of its sampling distribution (21.5) arises from the fact (which
we state without proof) that in normal samples, and only in normal samples,
sampling variations of the mean are completely independent of (and not
merely uncorrelated with) those of the variance.

There are other cases in which we find a quantity which is the ratio
of two independent variates, the numerator distributed like a mean and
the denominator like a standard deviation in normal samples. In such
cases, of course, the ratio t follows “ Student’s ” distribution. The most
important, perhaps, is that of regression coefficients.

21.24 Consider a linear regression equation —

(21.13)

where y, x are measured from their means and ^ is the parent value of the
regression. We will assume that for any fixed x the distribution of y is
normal as, for instance, is true if the joint distribution is normal. The
corresponding sample regression equation will be —

y^y =z h{x—x) . . , , . (21.14)

Then if are the sample variances of p; and y respectively it may

be shown that —

(»-^)SiV(w-2)

(21.15)

is distributed in “ Student’s ” form with v =n—2 degrees of freedom. The
result derives from the fact that is distributed like a mean in

normal samples whereas (s 2 ®-“&^ 5 i 2 ) /(n— 2) is distributed independently
like a variance. It is, in fact, an estimate of the variance of the residuals
of observed values about the regression line — cf. 9.24.

The expression for Hn (21.15) does not involve any of the parent para-
meters except ^ and consequently it may be used to test the significance
of irrespective of the other parameters.

tHE SAMPLING OF VARIABLES

493

Example 21.4. — In Table 13.1 (page 311) we gave some data for the
yields of wheat and potatoes m 48 English counties. The regression of
y (potato yield) on x (wheat yield) is found to be —

y-^e-0e5 = 0-0783 (^-15-791)

The value of the regression coefficient is small. Could it have arisen by
chance in a sample from a population for which /?=0 ?

We find—

b = 0-0783, V(^-2) = V46 = 6-7823,

^ 4-1749, 5^2 = 0-5340.

Hence, from (21.15) —

^ = 2-06, 1^=46

Appendix Table 4 does not carry us as far as v=46. For large v, t tends
to be distributed normally with zero mean and unit variance and a normal
deviate of 2-06 would be significant at the 5 per cent level but not at
the 2 per cent level. The regression is of doubtful significance.

More accurately, from the Fisher-Yates Tables we find the following
values of t for P=0-05 —

j;==40, 25 = 2-021; v= 60 25 = 2-000

and for P = 0-02 —

^=40, ^ = 2-423; = 60 25 = 2-390

This confirms our result that the observed t is significant at the 5 per
cent but not at the 2 per cent level.

21.25 We have remarked in 19.31 that the significance of a value of
Spearman's rank correlation-coefficient can be tested by the use of

Student's " distribution ; and we shall see later (21.34) that the product-
moment correlation can also be tested in the same way on the hypothesis
that there is no parent correlation. These facts are to be regarded as
mathematical accidents. They do not depend on the properties of
Student's " ^ as a ratio, but on the fact that the t~ distribution, being
a symmetrical unimodal distribution which tends to normality, may be
used as an approximation to other distributions of the same kind.

Fisher's distribution

21.26 Suppose that we have two samples, as in 21.21 with variances
and Then if the samples come from the same normal population
the distribution of the ratio' g = 5^/52 may be shown to be —

. ( 21 . 16 )

494

THEORY OF STATISTICS

where | may have any value from 0 to oo . This may be put in a rather
different form. In terms of the estimated variances Sj' and Sg', write —

, = = .... (21.17)

Then it may be shown that in normal samples from the same population
z is distributed according to the law

_ g V

where

J^2 =

(21.18)

(21.19)

As usual, we take so that the area of the curve is unity, and the
probability that we get a given value Zq or greater on random sampling
will be given by the area to the right of the ordinate at Zq,

21.27 This probability is not easy to tabulate owing to the fact that
it depends upon the two numbers and Fisher has therefore pre-
pared tables showing the 5 per cent and 1 per cent significance points of z,
and a further table of the 0* 1 per cent points has been given by Colcord and
Deming. These tables are reproduced by permission in Appendix Tables
6A, 6B and 6C. For practical purposes they are sufficient to enable the
significance of an observed value of z to be gauged. If the exact value of
the probability of obtaining a given value of z or greater is required, use
may sometimes be made of the tables of the incomplete beta-function.
Tables are also available for the values of the variance ratio itself
corresponding to specified probability levels. The quantity z of (21.17)
was used by Fisher instead of the ratio because linear inter-

polation is more accurate in the* 2 :- tables. The 5 per cent, 1 per cent
and 0*1 per cent points of the variance-ratio F are given in Appendix
Table 5.

Example 21.4. — Consider again the data of Example 21.3.

Here, as always, it is convenient to take the suffix 1 to refer to the
larger of the two estimates of variance.

We have —

o '2

^ '2
*2

37-875

5-4107

2 = 1 log«

5-4107

= 0-724

>’1=8, Va = 7

THE SAMPLING OF VARIABLES

495

From Appendix Table 6A we see that for these degrees of freedom the
5 per cent significance value of z is 0*6576. From Table 6B the 1 per cent
value is 0*9614.

The observed z lies between these two and is thus of rather doubtful
significance.

S '2

Alternatively F = = 4*25 and from Appendix Table 5 A and 5B

we see that the 5 per cent and 1 per cent points are 3*73 and 6*84, leading
to the same conclusion.

21.28 We shall consider this distribution and some of its uses in the
next chapter (Analysis of Variance). At this stage we may note that,
since it contains no unknown parameters, it provides a significance test
for the ratio of any two independent variates each of which is distributed
like a variance in normal samples. The distribution of a variance (or
equivalently, of course, of a standard deviation) is, in fact that of so
that z may be regarded as the distribution of the logarithm of the ratio
of two independent variates each of which is distributed as

Correlation coefficient in small samples

21.29 Although the distribution of the correlation coefficient in samples
from a bivariate normal population tends to the normal form as the
size of the sample increases, a fact which justifies the use of the standard
error for large n, the distribution diverges very remarkably from the
normal when n is small, and even when n is moderately large if the correla-
tion m the parent population is high. Further investigation is therefore
necessary before we can assess the significance of correlation coefficients
obtained from small samples.

21.30 The distribution of the correlation coefficient in samples from
a bivariate normal population was obtained in an exact form by R. A.
Fisher in 1915. Ordinates of the frequency-curves which give the
distribution have been worked out for various values of n and p, the
correlation in the population, and are tabulated in F. N. David's Tables
of the Correlation Coefficient. The general form of these curves is illustrated
in fig. 21,1, which shows the curves for p==-f 0*6 and various values of n.

A glance at this figure will show that even for a moderate value of p,
such as +0*6, the distribution of the coefficient is U-shaped for ^=3,
and, although unimodal, distinctly skew to the eye even for w =20. For high
values of p, such as +0*9, the distribution is skew for higher values of n.

As a result it is safe to say that the values of correlation coefficients
calculated from samples of less than five will throw no light on the existence
of correlation in the population. For samples of 20 or 30 we cannot
apply the standard error with much confidence if the correlation in the
population is likely to be very high, whether positive or negative. 50
seems to be the minimum number in the sample for the application of
the standard error if p is very Hgh, and 100 is safer.

496

THEORY OF STATISTICS

21.31 The equation giving the distribution of the correlation coefficient
is very complex, but Miss David's tables referred to above give the areas
under the frequency curves for various values of n, p and r. These tables
may be used to assess the significance of an observed value of r from a
bivariate normal population. For most practical purposes, however, use
may be made of a method due to R. A. Fisher, the essence of which is the
transformation of the distribution of r into a new distribution which is
approximately normal.

Fig, 21.1. — Frequency distribution of the correlation coefficient in samples from a
normal population with correlation +0-6 for various values of the number in the

sample n

In each case the total frequency, i.e. the area under the curve, is unity

21.32 Before we discuss this process, however, it is desirable to point
out the degree of applicability of our results.

(1) In the first place, it has been shown that the distribution of partial
correlation coefficients in samples of n is of the same form as that of total
correlation coefficients in samples of where p is the number of

secondary subscripts in the partial coefficient,

THE SAMPLING OF VARIABLES

497

(2) Secondly, our results are strictly true only for normal populations.
There is some experimental evidence to show that they are true for all
practical purposes even if the parent is moderately skew but remains of
the unimodal type ; but if there is any reason to suppose that the parent
is J- or U-shaped according to one or more variates, the student should
draw his conclusions with the utmost reserve.

Fisher’s transformation

21.33 If r and p are the correlations in the sample and the population
respectively, let us put

So that

r = tanh z p = tanh f

z* = i log,
C = i log.

1 + r

\-r

1+P

\-p)

( 21 . 20 )

Then it may be shown that z is, to a close approximation, distributed
normally about meail J with standard deviation
In fact, the mean of z is given by

' = 5 :+ 25 ^+ terms in etc.

and, for the ^r-distribution

. ( 21 . 21 )

. ( 21 . 22 )

. (21.23)

For say, is of the order of 0*001 even if p is high, which shows

how closely the ^-distribution lies to the symmetrical ; and is of the
order of 0-2, which shows that the distribution has nearly normal kurtosis.
In such a c^e z would difier from C W 0-05, which is not large, but
might be important in some cases. The standard error of z is, however,

and the factor -

-may, as a rule, be neglected in comparison.

VnS’ 2(w— 1)*

This is the basis of the statement above that z is normally distributed
about mean

We now give some examples of the use of the 5r-transformation in
testing the significance of an observed r.

This 4r is to be distinguished from the z of Fisher’s distribution of 21.26.

498

Theory of statistics

Example2l,b , — In Example 9.1, page 223, we found that the correlation
between the price indices of animal feeding-stuffs and home-grown oats
is 0-68, the sample consisting of 60 members.

This sample is large enough for us to use the standard error. If we do
so we get

l-(0*68)2

= 0*07 approximately

The correlation thus is undoubtedly significant.

We might, alternatively, use the z test, thus, to answer the question,
Could the observed value have arisen from an uncorrelated population? ”
On this hypothesis

p = 0 and f = 0

We have —

= 0-829

The standard error of z is —

V57

0-13.

The deviation of z from f is more than six times this, and we conclude
that our hypothesis was incorrect, i.e. that the population is correlated.

Example 21.6. — Continuing the previous example, could the observed
correlation have arisen from a population in which p=+0*8?

Here

^ = i log. [^ = 1-099
The deviation of z from f is, therefore,

1 •099-0*829 =0*270

This is about twice the standard error of z. It might arise, though
rarely, as a sampling fluctuation, and we conclude that p is likely to be less
than -f 0*8.

Example 21.7. — In Example 12.1, page 290, we found a partial correla-
tion of —0*73 (38 unions) between earnings of agricultural labourers and
the percentage of the population in receipt of relief, when the ratio of
numbers in receipt of outdoor relief to those relieved in the workhouse was
constant. Is this significant, and can it have arisen from a population in
which the real correlation is —0*667 ?

THE SAMPLING OF VARIABLES

499

Here

0- 27

1- 73

= -0-929

^ for an nncorrelated population = 0
-r A It 0-333

C if /5 == -0-667 = I log*

= -0-805

There is one secondary subscript in the partial correlation. Hence, the

standard error of z = v ^ --=:0*1715.

V 38 — 1 — 3

If ^=0, the deviation is more than five times the standard error and
is undoubtedly significant. If 0-667, the deviation is less than the
standard error and hence may very well have arisen from sampling
fluctuations.

Application of “ Student’s ” distribution to correlation coefficients
21.34 The test we have just given is of general application, but it is
worth noticing that if p=0, the distribution of the correlation coefficient
in small samples from a normal population may be tested by the '' Student''
distribution.

In fact, the distribution of the correlation coefficient assumes a par-
ticularly simple form for such uncorrelated populations, namely,

.... (21.24)

If we put

^ ~ • • • (21.25)

then it may be shown that t is distributed in the Student " form with
^^->2 degrees of freedom, and its significance may be tested accordingly.

SUMMARY

1, As an estimate of the mean of the population we may take the mean
of the sample, whether large or small.

2. If the mean of the population is known, we may take the mean
square deviation about that mean as an estimate of the variance of the
population ; i.e. the estimate is given by

s* =

500

THEORY OF STATISTICS

3. If the mean of the population is not known, a preferable estimate of
the population variance is the ** corrected variance of the sample, given by

4. This estimate is said to have n—\ degrees of freedom.

5. In samples from a normal population the parameter t, given by

^ x—m / — —

— 7-Vv+l
s

where v—n—l, is distributed according to the law (due to “ Student ”)

This distribution may be used to give the probability of getting a value
of t between specified limits on random sampling.

6. With two samples, x-^, , , , Xni and x^', . , . Xn'^, from the same
normal population, the parameter t defined by

I ^ 1^2

s' ^^+^2

where

is also distributed according to the above law, with v degrees of freedom.

7. With two samples, as before, with estimated variances

s'i =

s 1 s ^

the parameter 2 : = log — = ^ log

is distributed according to the law (due to R. A. Fisher)

y=yo 5+5

{Vie^^+Vz) 2

where

As usual, this distribution may be used to give the probability of
getting a value of z between specified limits on random sampling.
Alternatively tables are available for testing directly the ratio —

THE SAMPLING OF VARIABLES

501

8. The distribution of the correlation coef&cient in samples from a
normal bivanate population is not normal. However, putting

‘ ” 1

where p is the correlatiorr in the population, it may be shown that z is
approximately normally distributed about ^ with standard deviation

n being the number in the sample.

9. This result remains true of partial correlation coefficients, but in
the above formulae n must be taken to be the number in the sample less
the number of secondary subscripts in the coefficient tested.

10. In samples from an uncorrelated normal population the distribution
of r IS given by

The statistic t, defined by

is distributed in the “ Student ” form in such cases with n—2 degrees of
freedom.

EXERCISES

21.1 Find “ Student’s ” Hor the following variate values in a sample of
10: —6,— 4, —3,— 2,— 2, 0, 1, 1, 3, 5, taking m to be zero, and find from
the tables the probability of getting a value of t as great or greater on
random sampling from a normal population.

21.2 A farmer grows crops on two fields, A and B. On A he puts
worth of manure per acre and on B £2 worth. The net returns per acre,
exclusive of the cost of manure, on the two fields in five years are —

Year

Field A, £ per acre

Field B, £ per acre

16-5

I 18*5

i 19

j 22

502

THEORY OF STATISTICS

Other things being equal, discuss the question whether it is likely to
pay the farmer to continue the more expensive dressing. State clearly
the assumptions which you make.

21.3 The heights of six randomly chosen sailors are, in inches : 63, 65,
68, 69, 71 and 72. Those of ten randomly chosen soldiers are : 61, 62,
65, 66, 69, 69, 70, 71, 72 and 73. Discuss the light that these data throw
on the suggestion that soldiers are, on the average, taller than sailors.

21.4 In the data of Exercise 21.3, use the ^^-distribution to discuss
whether the samples can have come from populations which are identical
so far as height distribution is concerned.

21.5 In three samples of 50 lines each from Shakespeare's “ Romeo and
Juliet " (an early play), the following numbers of weak endings were
observed : 7, 9, 10. In three similar samples from “ Cymbeline " (late),
the numbers of weak endings were 15, 11, 12. Discuss the suggestion
that Shakespeare's prosody, as judged by the number of weak endings,
changed with advancing years.

21.6 A random sample of 15 from a normal population gives a correlation
coefficient of —0-5. Is this significant of the existence of correlation in
the population?

21.7 Show that in samples of four from an uncorrelated normal popula-
tion all values of the correlation coefficient are equally probable ; and that
for samples of less than four a zero coefficient is the most improbable.

21.8 What is the probability that a correlation coefficient of +0-75 or
less can arise in a sample of 30 from a normal population in which the
true correlation is +0 * 9 ? Compare this with the result given by assuming

the sampling distnbution normal with standard deviation

1 —
Vn

21.9 Test the significance of the partial correlation coefficients of
Example 12.1, page 290.

21.10 Show that in samples of 25 from an uncorrelated normal popula-
tion the chance is 1 in 100 that r is greater than about 0-43.

21.11 If two statistics both have the same dimensions show that their
ratio must be independent of the scale of the parent population. Hence
consider why “ Student's " t and Fisher's z (variance-ratio) are indepen-
dent of cr, the standard deviation of the normal parent.

21.12 By considerations similar to those of the previous exercise show
that in normal samples the distribution of the correlation coefficient
cannot contain either the parent means or the parent variances, but only
the parent correlation.

CHAPTER TWENTY-TWO

THE ANALYSIS OF VARIANCE

22.1 In this chapter we shall consider a technique of analysis which is
of wide application whenever samples of variate data can be classified
in groups. For instance, we may have a sample which consists of p
sub-samples, our interest lying in the question whether the total sample
may be regarded as homogeneous or alternatively whether there is some
indication that the sub-samples were drawn from different populations.
Again, we may have a number of plots of a cereal grown under diferent
manurial treatments. Our interest here is whether the manures exert
any differential ehect on yields ; and if we classify the yields into groups
according to the type of fertiliser applied we have the case, already
mentioned, of p sets of data which we require to test for homogeneity, p
being the number of different treatments. To take a more complex
case, we may have a number of observations taken by p different observers
each on a sample affected by q different effects, as for instance, if ^ labora-
tory assistants carry out an assay on samples of a drug from q diferent
suppliers. Our classification here is two-fold and we wish to discuss
whether there are any significant differences between the q sources of
supply and, independently if possthle, whether there are any diferences
between the results obtained by the p assistants.

In general we desire to answer the question whether some one variable,
treated as dependent variable, does or does not exhibit heterogeneity
when classified into arrays families ” or classes by one or more
independent variables.

A single independent variable

22.2 We shall discuss in the first instance the simplest case of a single
classification (i.e. according to one independent van able) and shall
proceed to the more complex cases later.

Suppose then that we have a set of variate- values divided into p families,
the number in the jih. family being %. We may array the values thus —

First family

Second family

^th family

^l*p> • *

503

. ( 22 . 1 )

504

THEORY OF STATISTICS

Let US denote by x., the mean of the whole set and by the mean of
the jth family. This is a new notation which will be very convenient
for later generalisation, a period replacing any subscript which is averaged.
Then, denoting by S summation over values of i from 1 to % and values

of j from 1 to we have the simple algebraic identity

)* = 'Z{x,i-Xj-\-x,-x )2

= 'L{Xii-x jY+2'L[x,,-x i){x ,-x )

V *3

+ ( 22 . 2 )

Now if we carry out summations over i alone we have

2^ (%-*.<)(*.#-*..) = {X.1-X. ) S {Xij-x ,) = 0

i t

since, by definition, Xj is the mean of % in the yth family. Hence we
have, from (22.2)

2 (Xti-xj* = s s

V *3 *3

. . (22.3)

*3 3

22.3 This is a fundamental identity and we pause to examine its meaning.
The expression on the left in (22.3) is the sum of squares of all values taken
about their mean, a quantity which we shall call the deviance. If the
total number of observations {=S%) is iV, the deviance is iV times the

variance of the total number of observations, and no confusion will arise
if we call it the total deviance.

The first term on the right in (22.3) is the sum of the deviances of each
family. Regarding the sum of squares of deviations from a mean as a
measure of variability, we may regard this term as expressing the variation
within families. On the other hand the last term on the right in (22.3)
is the sum of squares of means of families about the total mean and may be
regarded as expressing the variation between families. Thus we have
analysed the variation of the whole group into two parts, one expressing
variation within families, the other expressing variation from family to
family.

22#4 Strictly speaking, perhaps, we ought to call this process an analysis
of deviance, but it has become known as the analysis of variance. In
the particular case when ail families contain the same number n, (22.3)
simplifies in a way which exhibits how this term came into use. For then

S )2 = iV var X ~np var x

'Zn^{x f—x nll(x j—x

THE ANALYSIS OF VARIANCE

505

and hence, on substitution in (22.3)

= • • (22.4)

Now if we write s® for the variance of the whole, for the variance of
the p family means and for the variance within the/th family, we shall
have

s* = ls(s,»)+s„^ .... (22.5)

P 3

Our total variance is then expressed as the sum of two components, a
mean of the variances within families and the variance of the means of
families.

22.5 Equation (22.5) should be compared with equation (18.13) to which
it is formally equivalent. Our discussion of the sampling variation in
non-simple sampling was, in fact, a form of variance-analysis. The
effect of sampling from the parts of a patchy population is to increase
the variance by an amount equal to the variation of the means of patches
among themselves.

22.6 Now let us suppose that the p families from which our samples were
drawn are not different, i.e., that the data are homogeneous. Then the
variance of the whole sample will give us an estimate of the (common)
parent variance v. If N is large it makes no practical difference whether
we use the actual variance of the sample or the alternative estimate of
(21.3) obtained by dividing the deviance by iV— 1 ; but there are practical
as well as theoretical reasons for using (21.3) when the sample is small,
and we shall use it in all cases ; that is to say, we shall base our estimates
of the variance on the appropriate number of degrees of freedom (21.8).
An estimate of the parent variance v is then given by

.... ( 22 . 6 )

But this is not the only estimate we may derive from the data. On
our hypothesis as to homogeneity, the deviances within families provide
an estimate when divided by the appropriate number of degrees of freedom.
Thus a second estimate is given by

.... (22.7)

Finally, the means Xj are distributed with variance v jn^ in virtue of
(18.8) and it may be shown — we must omit the proof — that a third
estimate of v is given by

( 22 . 8 )

5o6

THEORY OF STATISTICS

22.7 Examination of (22.6), (22.7) and (22.8) will show that the various
numerators are the items entering into (22.3), while the degrees of freedom
forming the denominators are also additive, i.e. —

N^l = (N^p}+{P^1)

We may therefore exhibit our estimates of v in the form of a table as
follows :

TABLE 22.1. — Form of variance-analysis for a single independent variaMe

(1)

Deviances relating
to variation

(2)

Degrees of
freedom

(3)

Deviances

(4)

Estimates of v
(Column (3) divided
by column (2))

Between families

p-\

)

^l^jn,{x,-xy

Within families

N-p

N-pfj

Total

N^l

This convenient lay-out enables a check to be made in arithmetical
examples from the fact that in columns (2) and (3) the value at the foot
is the sum of values in the body of the table. This is not, however, true
of column (4).

22.8 Now suppose that we have carried out such an analysis for a
particular arithmetical case and derive three estimates v^, and of the
parent variance v. If these three values are in reasonably close agreement
we see no reason to reject the hypothesis that the families all come from
the same population, that the data are homogeneous, or that there are
no real differences between family means. On the other hand, if the
estimates are different (and significantly so in a sense we shall discuss
below) we may reject the hypothesis of homogeneity and conclude that
there exist real dif erences between some or ail of the families.

22.9 To make the argument satisfactory we require some criterion to
decide when the various estimates are significantly different. This brings
us to the second fundamental feature of variance-analysis. If the popula-
tion is normal the two estimates of variance derived from variation within
and between families are independent and their ratio is distributed,
independently of the actual value of the parent variance, in the form
of (21.16) and hence may be tested in Fisher's ;s:-distribution (21.26), or
the equivalent E- or variance-ratio distribution.

THE ANALYSIS OF VARIANCE

507

Note that neither estimate of variance can be independent of the
estimate derived from the total vanance, for the latter incorporates them
both. Our significance test must relate to the ratio of variation between
classes to variation within classes.

22.10 We shall not present a proof of the results stated in the 'previous
section but the following line of reasoning will indicate how such a proof
may be derived. For normal populations as we have stated, the mean is
distributed independently of the vanance (19.17). On the hypothesis
of homogeneity, the means of families are therefore independent of the
variances within famihes ; and consequently the estimate between families,
which is derived solely from the means, is independent of the estimate
within families, which is obtained by pooling the deviances within families.
Hence the fact of independence. That the estimates are distributed like
variances follow^s from an elaboration of the consideration that the mean
of normal samples is also normally distributed, so that the variance
between families is like a variance of a normal sample ; whereas the
variation within families is the sum of deviances and, like x^f is additive
in the sense that its total is distributed like a constant multiple of a
variance.

We proceed to consider two examples, one for large and one for small
samples.

Example 22.1. — The following table (from the Registrar-General's
Statistical Review of England and Wales for 1933, Part II) shows the
numbers of males married in England in that year classified according to
age and district. (Certain small numbers of unspecified age and those
under 21 have been omitted). Note the changes of interval at 25- and
35- years.

TABLE 22.2

District

21~

25-

Age (years)

30- 35-

45-

55 and
upwards

Totals

South-East

31,714

43,979

14,995

7.985

3,928

3,717

106,318

North

31,507

39,849

13,620

7,108

3,362

2,916

98,362

Midland

17,465

21,496

6,729

3,340

1,624

1,509

52,153

East

4,016

5,297

1,820

962

457

386

12,938

South-West .

4,323

6,065

2,218

1,177

514

580

14,877

Totals

89,025

116,676 39,382

20,572

9.885

9,108

284,648

The question we shall discuss is whether the average age at marriage
differs significantly between the different districts, i.e. we take district
as the independent variable. This, apart from its sociological interest,
might be an important point for decision if we were about to carry oat
a sampling inquiry into some quality which was related to age at marriage,
such as numbers of children per family.

5o8

THEORY OF STATISTICS

Taking the centres of the intervals to he 23, 27*5, 32 ‘5, 40, 50 and 57*5
years (the last being an approximation) we find —

TABLE 22.3

District

Mean age
(years)

Degrees of
freedom

Sum of
squares

Quotient, sum of
squares divided by
degrees of freedom

South-East

29-68

106,317

7,092,490

66-71

North

29 31

98,361

6,092,375

61-94

Midland

29-01

52,152

3,105,520

59-55

East

29-43

12,937

807,911

62-44

South-West

29-87

14,876

1,025,284

68-92

Value for the

whole area

29-43

284,643

18,143,921

63-74

This is not a table in the form of Table 22.1. It merely exhibits the
means and estimated variances for the different districts and the area
as a whole. We note that the differences between districts are not very
large but that the mean age at marriage is higher in the south than the
north. Is this significant in the sense that it could not be a sampling
effect such as would be obtained if the population were homogeneous ?

The sum of squares between classes is obtained as the sum of deviances
in the fourth column of the above table and Is 18,123,580. This is not
the sum shown at the foot, which is the deviance for the whole area
and is derived from the figures at the foot of the Table 22.2. The
difference between the two, 20,341, is the sum of squares between classes
as can be checked by direct calculation from the means.

We then find —

TABLE 22.4

Variation

Degrees of
freedom

Sum of squares

Quotient

Between districts

20,341

5085-25

Within districts

284,643

18,123,580

63-67

Totals

284,647

18,143,921

A test of significance is hardly necessary to show that the quotients
are in fact significantly different. But if we wish to apply the - 2 :-test
we proceed as follows —

THE ANALYSIS OF VARIANCE

509

We have

2 J log*

Vi =4. Vj r

5085-25

63-67

= 284,643

= 2-19

From Appendix Table 6C we have, for the 0*1 per cent points for = 4

(for V 2 =60) 0*8345

(for = 00 ) 0*7648

The observed value is far greater than these and hence is highly
significant. Alternatively F =5085 • 25 /63 • 67 =8 • 0 which again is beyond
the 0*1 per cent point (Appendix Table 5C). We conclude that the
differences in the mean ages between districts, though comparatively
small, are not accidental.

. Example 22.2. — Table 22.5 shows the yields of 30 plots of barley,
there beipg six plots of each of five varieties. In this table the independent
variable is the variety, so that rows and columns are interchanged as
compared with Table 22.2. Moreover the number of plots for each
variety is so small that we do not draw up a frequency distribution giving
the number of plots with yields between certain limits (on the principle
of Table 22.2) but simply the actual yields of the six plots. We are
interested in the question whether there is any significant difference in
the mean yields of the different varieties.

TABLE 22.5. — Yield of grain in grammes on plots of barley of one square yard, there
being five varieties and six plots of each
The tabular arrangement does not represent the physical lay-out of the plots
(Data quoted by Engledow and Yule, *‘The principles and practice of Yield Trials," 1926)

Plot

number

Variety

Mean

387

372

350

340

398

369-4

420

455

417

360

358

402*0

3 i

353

375

400

358

334 I

364*0

4 1

331

328

325

370

340

338*8

5 I

358

383

378

395

320 1

366*8

400

308

275

375

430

357*6

Mean

374*8

370*2

357*5

366*3

363*3 !

366*4

The mean of the whole is 366*4. The deviance is easily found to be 4|,934,
As in the calculation of a variance, we take some convenient working

510

THEORY OF STATISTICS

mean to simplify the calculation. Similarly we find for the contribution
between families, from the means of columns

*j * i

==6{(374-8 ~366*4)2+ . * .+(363-3~366*4)2}

= 1043

For the sum of squares within classes we merely subtract this quantity
from the total deviance. Our analysis of variance then becomes —

TABLE 22.6

Variation

Degrees of
freedom

Sum of
squares

Quotient

Between varieties .

1.043

260*75

Within varieties . 1

42.891

1,715*64

Total

43,934

We have here an interesting case in which the variance between
varieties is less than that within varieties. If this effect is real there
must be some negative intraclass correlation present, a point to which
we return below. To test the significance we have

1715*64

260*75

0*942

= 25, Fa == 4

From Appendix Tables 6A and 6B we see that, for these degrees of freedom
the 5 per cent point is 0*876 and the 1 per cent point 1*31. The observed
value lies between them and is just beyond the 5 per cent point. The
result thus is barely significant, i.e. the evidence is weak that there is
any real difference between the yields of the different varieties.

Some practical points

22.11 We proceed to consider a few practical points in the analysis and
interpretation of variance analysis in the case of a single independent
variable.

First of all, as regards the arithmetic. There is no difficulty about
determining the number of degrees of freedom, and the only arithmetical
labour arises from the determination of the sums of squares. The total
deviance is determined exactly as in the calculation of variance. We

THE ANALYSIS OF VARIANCE

5II

first find the mean, then, with a convenient working mean, determine the
sum of squares about that mean, and finally transfer to the real mean by
some such formula as

.... (22.9)

tj ij

which is only (6.4) in a different guise.

The next process is to determine the deviance between families. For
this we require the family means x Again with a v/orking mean if
desired (though, as in Example 22.2, it is not always necessary when there
are only a few families) we calculate the contribution 'Lnj{Xj—x^ )^.

A point to watch here is that each contribution to the sum is weighted
by the factor In the case where all the n’s are equal we have

'Ln[x j—x )2 = nYi{Xj --x
3 ’ j *

. . ( 22 . 10 )

The direct determination of the sum of squares within families is a
tedious business when the numbers in the families are large and ungrouped.
The required quantity can, however, be ascertained by subtraction as
in Example 22.2. This sacnfices a check on the arithmetic but is the
procedure usually followed.

In the hght of these comments the reader should venfy the arithmetic
of Example 22.2.

We might add that the formal analysis of variance does not relieve the
student from the necessity of looking at the data in a general way to
make a preliminary comparison. In Example 22.1 we tabulated the means
and remarked that they were not very different, even if significantly so.
Our work may be regarded as the simultaneous testing of the significance
of the differences between a set of means. Any pair of means can be
compared by the if-test ; we have tested all the differences together.

22.12 Consider now the application of the z-test. Strictly speaking,
this is valid only when the parent population is normal. There is some
evidence that in the contrary case the test remains valid provided that
the departure from normality is not great, as for instance, in a great deal
of biological material. But when the departure is considerable, special
measures may be necessary to deal with the significance test.

22.13 The reader will observe that the values tabulated in Appendix
Tables 6 are all positive, which implies (since x is a logarithm) that in
working out a variance ratio we always take the larger value for the
numerator. In Example 22.1 we examined the ratio given by (variance
between families) /(variance within families) whereas in Example 22.2 we
took the reciprocal of this ratio. The general rule is always to take the
larger figure as the numerator but this raises a point in connection with

512

THEORY OF STATISTICS

the significance test on which it is well to be clear. Our significance
values attached to a probability level of P per cent are chosen so that
there is probability P/100 that the values will be attained or exceeded.
The probability that a ratio will attain or exceed a given value k, or that
if it is less than unity its reciprocal will fall below 1 jk, is 2P /lOO, twice the
value for either contingency alone. When we are interested in either
contingency the probabihty levels given in the Tables should be doubled.

22,14 Appendix Table 6, will probably be sujSicient for most purposes
but it is worth recording that for large Vi and z is distributed approxi-
mately normally with mean — ^ variance ^ ^

Example 22.1, for instance, V 2 is so large that we may neglect its reciprocal
and, since v^—i the approximate result leads to the conclusion that z is
distributed normally with mean —0*125 and standard deviation 0*3535.
The actual value of 2* 19 deviates from the mean by more than six times
the standard deviation and is therefore highly significant. In our present
example the test is rough because is not large, but for and gi’eater
than 30 the approximation is quite good ; and even for lower values it is
useful to carry in one's head as a rough guide.

Relationship with intra-dass correlation

22.15 In 11,38 we considered the intra-class correlation of a number of
families. In the notation of the present chapter equation (11.33) can
be written

{l+{n-l)r} = nsl .

, ]L |!?£|=£!1

- n-l \ s* j

4 1

W — 1 5^

. ( 22 . 11 )

. ( 22 . 12 )

Mow s® is the variance of the total and is equal to S jnp where S is the total
deviance. Also is S-^jnp where is the sum of squares between
families. Writing Sg for the sum of squares within families (=S— Sj)'we
find from (22.12)

y ^ i ^2

5 n-l S

. (22.13)

This formula exhibits the relation between intra-class r and the con*
stitutent items of the analysis into sums of squares.

22.16 If now we denote by and the quotients obtained from Sj
and Sj we have

THE ANALYSIS OF VARIANCE

5^3

n — * ^2

Pin^l)

From (22ol3) we see that r is negative if and only if

S2>(n— l)Si

which is equivalent to

.... (22.14)

This condition was verified in Example 22.2. It is of rather rare
occurrence in practical cases.

Two independent variables

22.17 We now proceed to the case when the data are classified by two
qualities A and By p of one and q of the other, making pq sub-classes in
all. We shall consider in the first instance the simple case where there
is only one member in each sub-class. We shall denote the value of the
member in the ith class of A and the/th class of B by Xi^. We then have
the algebraic identity

s = S {(x,,-x, -Xj+JX )+(x,_-x )-\-{x,—x_ )}2

= S {x,f-x, -x,-{-x .)8+S (x, -x_)^+i: {x,-xy . (22.15)

y V ij

The product terms in the expansion vanish as in the case of the single
independent variables discussed in 22.2. This equation presents an
anatysis of the total sum of squares into three constituent sums. We
state without proof that if all the data are drawn from a population with
variance v the three items on the right are estimates oL(^— l)(y— l)v,
(/>— l)i; and (^—1) v respectively and they are independent each of the
other two. We may then present an analysis of variance in the following
form —

TABLE 22.7. — Form of analysis of variance for two independent variables with one
member in each sub-class

Variation

Degrees of
freedom

Sum of squares

Quotient

Between ^-classes .

p-i

Between B-classes .

q—l

If 1

Residual . '

Z(x,j—x,,-x,y+xy

— >3

Totals

Pq-l

S {Xiq'-X )*

514

THEORY OF STATISTICS

The first two items are obvious extensions of the variation between
families which we encountered in the one-way case of the single independent
variable* The item we have called residual in this table has no very
obvious interpretation but we may regard it as assignable to variation
within the sub-class. Each contributory deviation may be looked on
as the remainder when the effect of the classes A and B (if any) is removed
For instance is the deviation of the value from the mean of values

in the ith. 4-class. The mean of over the ^-classes is and

thus Xii-'Xi ) is the deviation from the average value obtained

by taking means for the 4- and 5-classes separately.

22.13 If the quotient for 4 is significantly different from the residual
quotient we may conclude that there is heterogeneity so far as concerns
4 ; and similarly for B, We now meet a new point which did not arise
in the case of a single independent variable. Suppose that the significance
tests show that the data are heterogeneous in 4. Can we then proceed
to test for heterogeneity in B ?

The answer in general is no, but there is one class of case in which it
is affirmative.

Suppose that the value % is made up of three independent and additive
parts

(1) the effect of belonging to the class 4^, say at.

(2) the effect of belonging to the class say

(3) a residual which is normally distributed with zero mean and
variance v.

Then we have

% .... ( 22 . 16 )

The reader should consider this hypothesis carefully. It is equivalent
to an assumption that the observations are affected by a systematic effect,
which varies from one 4-class to another but affects all B-classes alike
in the sub-class 4< ; a similar effect for B ; and the residual normal effect.

22.19 If niij is the population mean of a that of and so on we have
from (22.16)

^i. = ^ 1 +^.

Then

.... (22.17)

= . . (22,18)

THE ANALYSIS OF VARIANCE

53^5

the product term vanishing as usual. Now from (22.17) it is clear that
vanishes and hence the right-hand side of (22.18)
reduces to its last term. Thus the residual quotient is an estimatoi of the
variance which has just the same value as if and were non-existent.
That is to say, on the hypothesis represented by (22.16) the residual
quotient continues to offer an estimate of v, the variance of f.

It follows that, on this type of hypothesis, even if the ^-effects are
significant we can still test for the B-effects with the aid of the residual
quotient. We may also note that, in any case, if the are small, the
residual variance is not greatly afiected so that an approximate test can
be carried out.

Example 22.3. — The following is an example in which the dependent
variable is or may be subject to the influence of two independent variables.
Four varieties of potato are planted each on five plots of ground of the
same size and type ; and each variety is treated with five different fertilisers.
The yields in tons are as follows —

TABLE 22.8

Vanety

Fertiliser

1-9

2*2

2*6

1*8

2-5

1-9

2-3

2*6

2-2

1*7

1-9

2*2

2*0

2-1

1-8

2*5

2*3

2 4

We require to consider whether there is evidence that (^j) any difference
exists between the yields of varieties independently of the fertiliser and
(6) any differential effect is exerted by the fertiliser independently of the
variety.

Before carrying out an analysis let us look at the data generally. Since
each variety is treated once and only once with each fertiliser, we may
expect that comparisons of totals for the four varieties are permissible ;
the total yield of one variety is comparable with that of another because
they are both treated by the different fertilisers to the same extent.
Similarly, a comparison of fertiliser effects is legitimate because each
variety is equally represented in the five fertiliser totals. The data may
be said to be balanced.

It will simplify the arithmetic if we measure our yields about mean
2*0 and express them in tenths of a ton. Table 22.8 then becomes, on
the insertion of totals —

5i6

THEORY OF STATISTICS

TABLE 22.9

Variety

Fertiliser

Total

-1

-2

-1

-3

-1

-2

Totals

— 2

The sum of squares of yields (the 20 values in the main body of the table)
will be found to be 191. We then have

a;. =31/20 = 1*55

Nxl = 48-05

2 .)2 = s {x^)---Nx^= 191-48-05

i) %J

= 142*95

with (5x4)— 1 = 19 degrees of freedom.

We may now obtain the sum of squares between varieties direct from
the row totals of the table. These totals are, in fact, five time the means.
The sum of squares of means is thus 1 /25 of the sum of squares of row
totals ; but (and here is a slight trap) each square of a mean is to be
counted five times in ascertaining the sum of squares between varieties.
Thus the latter quantity is given by the sum of squares of row totals,
divided by five, less The sum of squares of row totals in Table

22.9 is 383 and thus the sum of squares between varieties is

383/5-48*05 = 28*55

with three degrees of freedom.

Similarly, the sum of squares of column totals is 377 and hence the sum
of squares between fertilisers is

377/4-48.05 = 46*2

with four degrees of freedom.

The analysis of variance then becomes — “

THE ANALYSIS OF VARIANCE

517

TABLE 22.10

Vanation

Degrees of
fieedom

Sums of
squares

Quotient

Between fertilisers

46*2

11-55

Between \ aneties

28-55

9 52

Residual

68*2

5-68

Totals

142-95

To test the effect between fertilisers we have

2 = i log, = 0-3545, = 4, = 12

This is not significant, being well below the 5 per cent point. Similarly,
for the effect between varieties

Q. CO

^ i Jrg = 0-2609

which again is not significant. A test of the variance-ratio direct leads
to the same conclusions.

We conclude that for these data there is no evidence of heterogeneity,
i.e. that they could have arisen from a population in which there was
no difference between the 5 delds of varieties and the fertilisers did not
differ in their effect.

Significance of the correlation ratio

22.20 At this point we turn aside from the development of the general
theory to show how the analysis of variance provides accurate tests of
significance for the correlation ratio, regression coefficients and the
multiple correlation coefficient.

The distribution of 9 ? ^ in samples from an unoorrelated normal population
ro'-y be derived from Fisher's j^-distribution. Hence we may test whether
an observed value of is significant of the existence of correlation in the
parent, assumed normal or approximately so.

When considering the correlation ratio in 11.6 we saw that for the
array of

wffiere

is the variance of the whole
a^x is the variance within arrays
is the variance of array means

5i8

THEOKY OF STATISTICS

If there are p arrays and % is the number of members in the yth array,
we may write this in the notation of the present chapter.

. . . (22.19)

Now let us regard the arrays as families or classes, and the items of the
arrays as class-members. Equation (22.19) is then an analysis of variance
in the following form :

TABLE 22.11

Vanation

Degrees of
freedom

Sums of squares

Quotients

Between classes

p-i

N<XlV^y

P-\

Within classes . . 1

N-p

N(rl[l-vpy)

N-p

Total .

N-l

In the last column we have anticipated results which are easily proved
as follows —

By definition,

Z{x,,^x == = Nul{\ - 77 /,)

Hence, 'Iinj{X j—x

Dividing the sums of squares by the appropriate number of degrees of
freedom, we get the results of the final column.

Now, if the population is normal and uncorrelated, the two quotients
are not significantly different ; for they are independent estimates of the
variance of x in the population, all arrays having the same mean and
standard deviation.^ We may test the significance of their difference by
the ar-distribution. We have —

^ = i log.

p-\ I N-p

= i log.

N-p

V,^p-l I

»'2 = ^-P 1

. ( 22 . 20 )
. ( 22 . 21 )

1 Strictly speaking, this is only approximately true of arrays of finite width. If the
ranges defining the arrays are very broad, the test must be used with reserve.

THE ANALYSIS OF VARIANCE

519

In equation (22.20) we have omitted the suffix %y in writing 17 Clearly
a similar test may be applied to f in this case referring to the number
of y-arrays.

22.21 From the relation (22.20) between z and it may be shown that
the distribution of 9; corresponding to that of ^ given by equation
(21.18), is

. . • ( 22 . 22 )

It will be seen that this involves the number p, i.e. depends on the
number of arrays into which the data are grouped. This fact is importantj

{ — -wS

and reveals that the use of the standard error given in 19.27, can

be no more than an approximation at the best ; for that formula does not
contain p.

22.22 It is interesting to note that, since rj'^ is positive, its mean value
will not be zero. The mean value (which differs from the square of the
mean value of 7) is given by

(?) = . . . . (22.23)

Example 22.4. — Let us consider the data of Table 9.3 (correlation
between stature of father and stature of son), in which 7a;v=9;^^aj=0*52.
We know that the distribution is approximately normal, a fact which is
borne out by the approximate equality of the two correlation ratios, and
hence we may appty the foregoing theory with considerable confidence.

We have, for —

= ^— 1 — 16

= El-p = 1078-17 = 1061

= i log<

{ 0 - 52)2
1 -( 0 - 52)2

1061

= 1-60

From Appendix Table 6C we see that the 0-1 per cent significance
points are as follows —

= 12 1^1 = 24

^ 2=60 0-5992 0-4955

0-5044 0-3786

The observed z is therefore very strongly significant of correlation in
the population.

Test of linearity of regression

22.23 In 11.7 we saw that the regression of y on x was linear if, and
onl}?' if, r^=0. An important question to decide is, therefore, can
an observed value of have arisen from a population in which the

regression is linear, i.e. the true value is zero ?

520

THEORY OF STATISTICS

This question can be decided by the 2 :-test in a similar manner to that
of 22.20 and 22.21. We consider the analysis of the sums of squares of
deviations from the regression line into two parts : ( 1 ) deviations within
arrays, and ( 2 ) deviations of means of arrays from the regression line. In
this way it may be shown that the linearity may be tested by taking

^2 y2 A

Vi—f -2 I .... ( 22 . 25 )

v^ = N-p 1

Example 22.5. — In considering the correlation between old age,
pauperism {x) and the proportion of out-relief (y), Yule found {Economic
Journal, 1896, 6 , 613)

iV =235

r = + 0-34
Tl^y = 0*46

= 0*39

for a grouping of 19 ^-arrays and 8 y-arrays. Can the regressions be
supposed liaear ?

For the ^if-arrays, N--p = 216, ^—2 = 17

^ ( 0 - 46 ) 2 -( 0 - 34)2
1 -{ 0 * 46)2

= 0*12177

log, ^0 • 12177 X^)

= 0-218

The 5 per cent point for Fi=17, ^ 2 == «>, is about 0*25, and there is thus
no reason to suppose from the observed z that the regression is not linear.
Alternatively for the variance ratio F we find

(216x0-12177) ^^,55

For the jy-arrays, similarly, ^—2 = 6 .

^( 0 * 39 ) 2 -

2 = i log.
= 0-244

-(0-34)2 227>

l-(0-39)*>

This also will be foimd to lie within the sampling limits, and the test
therefore does not reject the linearity of either regression.

THE ANALYSIS OF VARIANCE

521

Significance of the multiple correlation coefficient

22.24 The multiple correlation coefficient is in many ways analogous
to the correlation ratio, and we may test its significance by a procedure
very similar to that used for the significance of the correlation ratio and
regressions.

Consider the regression equation with p variates,

% = & 2^2 + & 8 ^ 3 + • • •

the variates being measured from their means.

We may regard the deviations of observed values of as composed of
two parts : (1) deviations from the values of given by the regression
equation, and (2) deviations of the latter from the mean of x^. The sum
of squares can be analysed accordingly.

The sum of squares of deviations of observed values of % from the
mean of by definition, and has N—l degrees of freedom.

The sum of squares of deviations of observed x^s from the regression
values is ^ ^ which, by the definition of . . . 3>), is equal to
iVai2(l— i?f(2 . . . This has N—p degrees of freedom, for has
iV— 1 degrees of freedom, of 2 has N—2 degrees, and so on. Writing
R for jRi (2 we may express the analysis in the following tabular

form : —

TABLE 22.12

Variation

Degrees of
freedom

Sums of squares

Quotients

Between classes

p--\

RWo-i*

1 —

(Regression values from
mean.)

Within classes .

N~p

(Deviations from regress-
sion values.)

Total

N-1

Now if the parent value of R is zero, the quotients should not differ ^
significantly; for and 63^2+ • • • then uncorrelated, and

hence deviations of x from the regression values are uncorrelated with,
and independent of, deviations of the regression values from the mean,
the population being normal.

Hence we may test the significance of R by putting

^ i loge

R^ N-p

v^^N--p

(22.26)

(22.27)

522

THEORY OF STATISTICS

It will be seen that equation (22.24) is of the same form as equation
(22.20). The distributions of and 7j^ are formally identical, and we
have, for instance, corresponding to equation (22.23),

(i?2)

-±d

N-\

. (22.28)

Example 22.6. — In Example 12.3, page 299, we found i?j(23)=0-74.
Is this significant ?

We have —

^ = 3, iV = 38

= 2 , ~

2 i log.

/ (0-74)® 35\

\l-(0-74)2‘ 2 )

= 1-53

For j^i=2, the 0*1 per cent significance points are —

- 1^2 =30 1*0859

1^2 =: 40 1-0552

The observed z is well above these values and hence R is significant.
Unequal numbers in classes

22.25 The treatment given in 22.16 to the case of two independent
variates was based on the assumption that there was only one member in
each sub-class. In the contrary case an accurate treatment is much more
difficult and we shall not be able to deal with it here. The following
remarks are intended as a preliminary to further reading —

(a) If the number in each sub-class is the same the foregoing theory
still applies.

[h) The theory also applies if the numbers in sub-classes are propor-
tionate, that is to say, if the frequency in the sub-class .4^ is a
constant multiple of (A^) {B^) where {A’) and {B,) are the frequencies
in the classes and Bj respectively.

(c) In other cases the theory does not apply ; but if the numbers in
sub-classes are not very different from equality or proportionality,
an analysis carried out on the means of sub-classes as if they were
the primary data, one to each sub-class, will probably not be
misleading, although it sacrifices some information.

(d) In any case di pxq classification with more than one member in
the sub-classes can always be regarded as a one-way classification

THE ANALYSIS OF VARIANCE 523

into pq classes. An analysis on these lines will provide a test of
homogeneity but does not distinguish, as it were, whether
departures from homogeneity are due to A or B or to a mixture
of both.

Non-normal variation

22.26 Some comments are also desirable, though again the matter is
too complicated for detailed treatment, on the assumptions of normality
which underlie the exact treatment of significance tests in the analysis
of variance. When the parent population is not normal estimates of
means are not independent of variances, so that the quotients given by
the analysis are dependent. Further, the logarithm of the variance-
ratio is no longer distributed in the 2 -form. We have already referred
to the fact that sampling and theoretical inquiries suggest that if deviations
from normality are only moderate, the theory still applies as an approxima-
tion. Sometimes the variate may be transformed so as to bring it nearer
to normality or the variances in the different classes nearer to equality.
In certain cases, by a process of randomisation before the data are collected,
it may be ensured that the 2 -test remains valid even where the parent is
not normal, though this amounts to a change in the nature of the inference.
These topics, however, are outside the scope of this book.

The case of three independent variables

22.27 The results appropriate to two independent variables may be
extended. The general case of n independent variables is rather com-
plicated and indeed data so completely specified for n greater than three
are rare. We shall conclude this chapter by stating without proof the
results for three independent variables, commenting on one or two new
points, and giving an example.

Consider then the case where there are three classifications into 4-,
B- and C- classes, one member in each sub-class typified by With
an obvious generalisation of previous results we have (summation extend-
ing over all t, j, k)

+ S(%. -sc*..—

+ .+3;...)®

(22.29)

the summations extending over all members of the sample, say pqr m
number, where there are p -classes, q B-classes and r C-classes.

524

THEORY OF STATISTICS

Each item on the right in (22.29) provides an estimate of the parent
variance on the hypothesis of homogeneity. The first three items are
of the type between classes which we have already encountered.
The next three are known as interaction terms. The last is a residual and
may also be regarded as an interaction of second order. We have then
an analysis in the following form.

TABLE 22.13. — Form of analysis of variance for three independent variables with one
member in each sub-class

Vanation

Degrees of
freedom

Sums of squares

Residual

Between ^-classes .

„ B-classes .

„ C-classes .

Interaction A B
„ BC

„ CA

Residual

p-i

q-l

r-1

{p-l){q-l)

(r-l)

- a : )»

'^Kk-x..k-x,,+x„y

^{Xtjk+X,..+X_,_+X„l,

-X,}-X,k-X,ji-Xj^

The quotient
of the sum of
squares by the
corresponding
number of
degrees of
freedom

Totals

pqr-1

22.28 As in 22.18 and 22 . 19 , if the variate is regarded as the sum
of three class effeets a^, and and a normal residual the residual
quotient continues to provide an estimate of the variance of It is
therefore customary to test the quotients between classes in relation to
the residual quotient.

We also have, however, three interaction quotients which, on the
hypothesis of homogeneity, should also be equal, within sampling limits,
to the residual quotient. If the interaction quotient AB is not equal,
within such limits, to the residual we must reject the hypothesis that the
variation can be expressed as the sum of the two class effects and b^.
The class effects are, so to speak, entangled, or they interact."' Similarly
for the other two interactions.

Example 22,7. — The following example typifies a situation of fairly
general occurrence but has been simplified somewhat to reduce the
arithmetic. Suppose we have two manurial treatments which we wish
to test. We will suppose that they are each applied to five varieties of a
cereal, and that, to give the expenment greater generality, it is repeated
at four different stations. Our 40 yields are then classified into a 4 X 5 X 2
grouping, four stations, five varieties and two treatments. We will
suppose that the yields, measured about some convenient working mean,
and expressed in some convenient unit, are as given in Table 22.14, wherein
Tx and T | refer to the two treatments.

THE ANALYSIS OF VARIANCE

525

TABLE 22.14

Stations

Vaneties

Totals

T T

'x,

T, " T.

T. " T,

-6

-4

-7

-3

-5

-22

-18

-2

-1

-1 -3

-4

-1

—2

-16

- 6

-^2

3 -4

2 6

-1

-2

-8

-4

— 17

The sum of squares of the 40 values in the main body of the table will
be found to be 640. Thus we have

^ = -8/40 = ~-0-20 ^

Nxi = 1-6

= 638-4

Now we find the sum of squares between stations (S), varieties {V),
and treatments separately. The yields for the four stations are the
totals of the two columns on the right in Table 22.14, namely, -40, -22,
13, 41. The sum of squares of these values is 3934. Now (the first
suffix referring to S)

S = S . . (6)

In the column totals there are 5x2=10 members contributing to the
sum ; but the summation on the right in {h) takes place over the four
stations and the 5x2=10 members for each station. Thus

i,j,k %

where the y’s are the totals,
Thiis

= 393-4

S = 393-4-1 -6

= 391 - 5 .

and this gives the sum of squares between stations.

{«)

526

tHEORY OF STATISTICS

Generally, if we require the sum of squares between ^-classes in a
p X q x r classification we have

♦ h W «

The five totals of varieties are 1, —6, —19, —4, 20 with a sum of squares
equal to 814. Thus for the sum of squares between varieties we have

^-1-6 = 100-15 . . . (d)

We leave the student to check as an exercise that the sum of squares
between treatments is 16*9. . . . . . {e)

Now we have to find the interaction terms. For this purpose it is
most convenient to condense the primary Table 22.14 into three others,
of which we will write down one. If we add the yields for the two treat-
ments on any particular variety and station, we obtain the following —

Stations

Vaneties

2 3

Totals

-10

- 6

-17

- 8

-40

- 3

- 6

- 7

- 5

•

- 1

-22

- 4

Totals

- 6

-19

- 4

The sum of squares of values in the main body of the table will be found
to be 1112, Each entry is the sum of two values and, with an obvious
extension of previous results we have

1112

S )2 = ii^^l.6 --=554*4 . . (/)

Now for the interaction 5 F we have

-X , +X -X. )^—1,{x,-x

Substituting from (/), (c) and (d) we have on the right

554-4 -391 -8-100- 15

= 62-45 .... (g)

which is the required interaction sum of squares for S V.

THE ANALYSIS OF VARIANCE

527

Again we leave the student to calculate the other two interactions to
obtain that for VT as 3*85 and that for TS as 2* 10. We have finally
(the residual sum of squares being calculated by subtracting the sum of
the other terms from the total deviance) —

TABLE 22.15. — ^Analysis of variance of Table 22.14

Vanahon

Degrees of
freedom

Sum of
squares

Quotient

Between stations (S)

391*80

13C-60

Between varieties

100-15

25-04

Between treatments (T) .

16-90

Interaction SF

62-45

5-20

.. VT

4 j

3-85

0-96

ST . ,

3 >

2-iO

0-70

Residual

61-15

5-10

Total

638*40

Now we first of all test our interactions against the residual term with
a quotient of 5*10 and 12 degrees of freedom. We find in fact that
they are not significant to a 5 per cent level. This implies that we may
assume that there is no “ entanglement between the factors and that
there is support for the hypothesis that the three are affecting yields
independently. We can then turn to a consideration of the main effects.

We find that the differences between stations are highly significant,
those between varieties are not significant at a 1 per cent level bat are
so at a 5 per cent level, and that differences between treatments are not
significant. We conclude that the variation in yields is due to variation
between stations and (perhaps) between varieties, but cannot be ascribed
to real differential effects between treatments without further inquiry.

SUMMARY

1, The analysis of variance is essentially a procedure ior testing the
differences between different groups of data for homogeneity.

2, For a single independent variable (classification into groups according
to one quality) an analysis may be carried out to show estimates of the
variance between and within classes whether the class-numbers are equal
or not. Homogeneity may be tested by comparing the estimates,

3, For small samples and normal parent variation the ratio of between-
and within-class variance may be tested in Fisher’s jsf-distribution.

528

THEORY OF STATISTICS

4. For classification according to more than one quality a more elaborate
form of analysis may be employed. The method applies only when the
numbers in sub-classes are equal (or more generally, proportionate) but
is probably a fair approximation when they are near equality.

5. The exact test of significance does not apply to non-normal variation
except as an approximation, but where departure from normality is not
great, the approximation is probably fair.

6. The analysis of variance provides exact tests of significance (in the
case of normal variation) for the correlation ratio, departure from linearity
of regression, and the multiple correlation coefficient.

EXERCISES

22.1 The following shows the lives in hours of four batches of electric
lamps —

Batch 1 : 1600, 1610, 1650, 1680, 1700, 1720, 1800

Batch 2 : 1580, 1640, 1640, 1700, 1750

Batch 3 : 1460, 1550, 1600, 1620, 1640, 1660, 1740, 1820 *

Batch 4 : 1510, 1520, 1530, 1570, 1600, 1680.

Perform an analysis of variance on these data and show that a significance
test does not reject their homogeneity.

22.2 Considering two samples as two families of values, derive an explicit
form for the ratio of estimated variances between and within families
and hence derive the ^-test for the difference of means in normal samples
with equal variances as given in 21.21. (The distribution of the variance-
ratio for Fi=l reduces to that of t^).

22.3 Four experimenters determine the moisture content of samples of
a powder, each man taking a sample from each of six consignm^^nts. Their
assessments are —

Observer

Consignment

3 4

Perform an analysis of variance on these data and discuss whether there
is any significant difference between consignments or between observers.

THfe ANALYSIS OF VARIANCE

529

22.4 Verify the arithmetic and the significance tests of Example 22.7.

22.5 Test the significance of the two multiple correlation coefficients of
Example 12.3, page 299, other than the one tested in Example 22.6.

22.6 Test the linearity of the regression of the distribution of cows of
Table 9.4, page 204 (referring to Exercise 13.1).

22.7 Examine how, in the analysis of variance, sums of squares between
classes may be regarded as interactions of zero order and (in the case
of three independent variables) the residual may be regarded as an
interaction of the second order.

22.8 (Data from Mahalanobis, /. R. Statist, Soc., 1946, 109, 325). The
following table shows estimates of an index of the cost of living in an
area of Bengal in 1945 made by five investigators each working in each of
five areas.

Investigator

Area

270

263

264

263

260

280

265

274

279

275

284

278

271

296

271

269

272

297

274

279

267

269

263

284

Perform an analysis of variance to see whether there are significant
differences between areas and between investigators.

CHAPTER TWENTY-THREE

SOME PROBLEMS OF PRACTICAL SAMPLING

23.1 In the previous seven chapters we have discussed the interpretation
of samples and developed various branches of theory which are designed
to give precision, in the sense of the theory of probability, to inferences
drawn from the sample to the population. At the outset (Chapter 16)
we considered briefly the types of sampling to which our theory is
applicable, noting in particular the fundamental importance of randomness
in the selection of data. We shall now examine in more detail some of the
problems arising in the selection of samples to which our theory may apply.

23.2 The complete process of sampling consists in effect of three stages,
there being considerable scope for judgment at each stage.

(1) If there is no natural unit, and often even if there is, we have to
decide what shall be our unit for the purposes of sampling. If our problem
is, for example, to determine the mean yield per acre of a certain crop
over a certain large area, there is no natural unit of area over which the
3 deld can be measured at each of n points in the large area. We must
therefore fall back on practical considerations to decide whether our
sampling unit shall be something very small, say a square yard, something
a good deal bigger, say 1 /10th acre, or something larger still, such as an
acre or more. If, on the other hand, the problem is to estimate by way
of sampling the proportion of a certain human population possessing a
certain characteristic, such as blue eyes, or surname beginning with H,
or age under 21, the natural unit is the person ; but this, as we shall see
presently, is not necessarily the most convenient unit for sampling
purposes.

(2) The unit having been fixed, the next step is to decide what shall be
the process of sampling : if it is agreed that the process should be a
random one, how is this randomness best secured ? If it appears possible
that some departure from unrestricted random sampling may lessen the
cost, or may even lower the standard error of estimation, what then shall
be the procedure and will this procedure carry with it any countervailing
risks ? How are we to treat the cases in which a member that we intended
to include cannot be found or, if found, will not provide a reply ?

(3) The sample having been taken, i.e. the specific units to be included
in the sample having been determined, the final stage of the work is the
measurement, description, or (to use the term in a very general sense)

SOME PROBLEMS OF PRACTICAL SAMPLING

what we may call the examination of the units included in the sample.
Properly speaking, this is no part at ail of the sampling process m the
narrower sense ; that was completed when we had determined which
specific members of the population were to be included in the sample.
Examination of the units is a process of observation such as we would
have had to carry out even if we had decided to deal with the entire
population and not a mere sample. But it is a process fundamental to
our work and must be considered here, for careless or incompetent
examination '' may lead to the most serious, and sometimes astonishing,
errors.

We will consider these three stages in the order given, as this will couple
the work of the present chapter most closely and logically with that of
the preceding chapters.

Size of the sampling unit
Example 23.1. — Effect of size of unit on has
We take, first of all, an example illustrating the importance of the
sampling unit in some types of inquiry. In an investigation into the
yield of jute in Bengal in 1940-41 (Mahalanobis, /. Roy. StaL Soc., 1946,
109, 325) material was collected for five different sizes of sample-cut from
the fields, ranging from one square foot to 256 square feet. In each
field (which was selected at random) an area of 16x16 feet w^as chosen,
also at random, and the crop was harvested in a number of sub-cuts
supplying yield rates for the sizes : 1 x 1, 3 X 3, 12 x4, 12 x 12 and 16 x 16
feet, the latter being the whole plot. The following are the estimates
of the yield in lb. per acre based on the various plot sizes —

Size (ft.) Estimated yield (lb. per acre)
lx 1 27,271

3x 3 17,462

12 X 4 16,080

12x12 16,763

16x16 16,828

Evidently the estimates based on the two smallest sizes of plot are
seriously biased. In this particular case it was easily shown that the
differences could not have been sampling effects.

The reason for this effect is not yet beyond doubt, but apparently it
is due to unconscious bias on the part of the observer, who, in measuring
out the plot, has a tendency to include rather than to exclude plants on
land near the boundary. This effect naturally diminishes in proportion
as the plot becomes larger. The remedy in this case is clear ; it is simply
not to use plots which are too small.

23.3 For all practical purposes the case we have just considered may be
regarded as one in which the area covered is continuous, so that there
is no unit indicated by the nature of the data. We could, it is true,

532

THEORY OF STATISTICS

regard the individual plant as the ultimate unit ; but for practical reasons
we cannot, in an extensive inquiry, bother ourselves with the selection
of plants. We must select fairly large areas, and the question then
arises how the size of those areas is to be determined. In Example 23.1
the bias appearing for very small areas dictated a lower limit to the proper
size but did not suggest an upper limit.

23.4 Even for discontinuous units the same type of question can arise.
Suppose, for example, we are sampling a country for the purpose of
determining the size of population or some similar demographic character-
istic such as would be given by a census. I'be ultimate unit is the indi-
vidual human being, but it may be very troublesome to pick out individuals
at random. Shall we lose anything by sampling with families as units,
or houses, or streets, or blocks or even whole wards ? Again, in an
agricultural inquiry, do w^e lose anything by taking as our unit the farm
instead of the individual field ?

23.5 Such questions rarely admit of a simple answer. In general there
will be a group of considerations in favour of choosing as large a unit as
possible and another group in favour of choosing a small one. Among
those of the first kind we may mention economy (e.g. because less time
and travelling are involved if the individuals are grouped and have to
be visited, or because information has already been tabulated for the
larger units). Among those of the second are the desirability of not
clustering sample-members too closely when the population is thought
to be patchy Additional complications may arise when our units "
are of different sizes, such as farms, for then there is some intuitive ground
for feeling that the different units ought to be given varying weights.
When the sizes of the units are known we can sometimes deal with the
problem as one of stratification, which we consider below, but there are
some rather complicated points arising in this branch of the subject
which have not yet been completely solved.

Some sampling procedures

23.6 We shall now consider some sampling procedures which depend
for their efficacy on prior knowledge of the population. When nothing
is known about the population a purely random selection of members
is the best. It avoids bias and can be made to provide information about
the standard errors of the quantities under estimate. Only rarely,
however, do we embark on ,an inquiry in complete ignorance about the
parent population. Our knowledge may be only vague and general, but
even so we can often apply it to improve the precision of our estimates.
Moreover, it is often highly inconvenient and expensive to draw a purely
random sample from a large existent population (e.g. by the use of random
sampling numbers) and practical necessity may dictate a modification of
the random process even though no theoretical gain in accuracy or
precision may result.

SOME PROBLEMS OF PRACTICAL SAMPLING 533

Stratified sampling

23.7 We referred briefly in 16.39 to the process of stratification, in which
we divide the population into strata and draw a random sample of
specified size from each stratum. Sometimes our stratification may be
a purely geographical basis, as for example if, m sampling farms from
England, we decide to draw a certain proportion from each individual
county. Sometimes it may be by reference to a variate-value, as when
we decide to draw certain numbers of farms in certain size groups irrespec-
tive of their geographical position. The operation of stratification may be
undertaken either to improve the value of an estimate or merely for
administrative convenience. If the strata are determined by some

natural '' factor the sampling process by stratification will also facilitate
comparison of the strata among themselves, which may be a subsidiary
object of the inquiry.

Sampling fractions

23.8 Suppose we have a population stratified into k strata, the number
in the ith stratum being Ni and the total number (2(iVj)) being N. We
take a sample of n members such that the number chosen from the
stratum is Suppose that we desire to estimate the mean value a of
a variate x in the whole population. How shall we choose the numbers

fti ?

We shall assume that if % is the jth member of the sample of the
estimate is of the form

k Hi

S S .... (23.1)

♦«»i

where the A's are constants to be determined. This assumption may be
expressed by sa 5 dng that we are looking for a linear estimate. Among
all the possible estimating functions of this kind we shall seek the one
which has the smallest variance. There are obvious advantages in an
estimate with the minimum of sampling fluctuation.

If the mean value of % in the ^‘th stratum is we have

1 *

a == S NiOCi .... (23.2)

Thus, writing E to denote the taking of a mean value we have, from
(23.1) and (23.2)

£j S (A,,%) I .... (23.3)

and since, by definition £(%)=5=a| we have

534

THEORY OF STATISTICS

If this is to be generally true independently of particular values of
we must have

ni N

2 A,, = . . . . (23.5)

j=i

This provides a first condition on the A's in order that the estimate may
have the true value as its mean value — that it should be unbiased in a
sense we define in 23.17. If A*- is the mean of A^^ in the ith set we may
write this as

- n.N •

(23.6)

Now consider the condition that the variance of t shall be a minimum.
Since E denotes a mean value we have for the rth stratum

var .

ir — £ I^E I ^ } J

This is equal to

E^Xi,{x„-ay-\- S' X,j\i{X(j-cc,){x,i-a,)^

where E' denotes summation over values of; and I except those for which
If the variance of in the ith stratum is this is equal to

S A|c;?+S'{A,-,A,fE(sr„— a.)(A;,j-a.)} . . . (23.7)

3 3*^

Now since there are 1) values for which y=j=/

1 r f ® -|

E[Xij-cit){x,i-a.,) (iVf-l) LiS

S (%-aj)*

N,~l

(23.8)

SOME PROBLEMS OF PRACTICAL SAMPLING

535

Fiom (23.7) and (23.8) we then have

varS(A,A,) =2A„a.^-S' A„A.,^

} i s.i J^i — i

-i\r,-SA*.-«?Ai

)8+2iV,SA„A<.

AT.-l

iV,S(A,,-A.)«

+M.(iv,— %)A, 2-

(23.9)

Now t is the sum of k items, each of which comes from a different stratum
and is therefore independent of the others. Consequently the variance
of t is the sum of the constituent variances, i.e. is the sum over i of the
expression on the right in (23.9). This is dearly a minimum if, for all i

= 0 . . . . (23.10)

This is equivalent to saying that within any substratum the A's must be
equal, which is what we should expect, for there is no reason why one
should be greater than another.

We then have

var t

N^-l

«,A1

1 ^

~ m i N,-i fit

1 N

+ constat

We have to minimise this for variations in subject to
YkTii ^ n ^ constant

(23.11)

(23.12)

It may easily be shown by the use of differential calculus that the minimal
values of are given by

N,-l'

. ( 23 . 13 )

536

THEORY OF STATISTICS

If now Ni is large we have approximately

nf a (23.14)

« a< . . . . (23.15)

Thus the ratio which the sample-number bears to the stratum number
Ni varies as the standard deviation of the stratum.

23.9 This interesting result has some important applications in stratified
sampling. We need not consider the case in which the a’s are known
exactly (for we should rarely have this knowledge without knowing the
means, in which case we should not be estimating the mean of the whole
from the sample). There remain, however, two classes of case where the
result is useful ; when —

(a) The standard deviations are known approximately from prior
information. In such a case we can determine the a*s from (23.15) to
some degree of approximation. An estimate based on a sample obtained
in this way, though not perhaps as good as it might be, will at least be
better than if we had ignored our knowledge of the standard deviations.

(b) A pilot inquiry on a small scale can be conducted to determine the
standard deviations approximately. This wiU bring us back to case (a).

Example 23.2. — (Data from Yates, J. Roy. Slat. Soc., 1946, 109, 12).

The Farm Survey of England and Wales covered all holdings of five
acres or more. Prior information was available as to the size-distribution
of these holdings as follows —

Size group (acres) Number of holdings

5 and less than

101,450

25 „ „

100

111,360

100 „ „ „

300

65,210

300 „ „

700

11,150

700 and over

1,430

290,600

We wish to take a sample, say, of about one in seven, or about 40,000
holdings, in order to estimate some factor foi the population of farms
such as the arable acreage. What fractions of the various size groups
should we choose ?

If we have, in the general case, a sample number in the ^’th stratum,
where S{r^)==5?^, we shall take as our estimator of the mean of the whole
population the statistic

SOME PROBLEMS OF PRACTICAL SAMPLING

537

where denotes the mean of the sample values from the ith stratum.
This is an unbiased estimator in the sense of 23.17 for the mean value
of Xi is the same as the mean value of over the iih stratum, i.e is
Furthermore, the variance of x will be given by

var X =11

) var A,.

(N,yN~r, ^

{nJ r.

(?-)

approximately

The reader may verify as an exercise that when is equal to Hi as given
by (23.15) this reduces to the minimal variance given by (23.11) to our
degiee of approximation, which is reached by writing instead if — 1
in the denominator.

We do not know the standard deviations of the factor under investiga-
tion in the various strata but we may make some very plausible assump-
tions. There must clearly be some high correlation between arable
acreage and farm 'area. Let us then suppose that the variability of the
one is proportional to that oi the other, i.e. that our sampling fraction can
be taken as proportional to the standard deviations of size of farm. A
sketch of the histogram of the data vdll show that the distribution is
approximately J-shaped. If in any stratum the farms were distributed
equally frequently with respect to size (i.e. if the histogram were actually
the frequency distribution) the variance of a stratum of width h would be
A^/12 and hence its standard deviation would be proportional to k. Let
us then choose our sampling fractions proportional to the widths of the
size groups.

The last group, 700 acres and over, has an un=ipecified upper limit. We
will, theiefore, suppose the standard deviation very large and sample
100 per cent. The range of the other groups are 20, 75, 200 and 400
acres and thus our fractions are proportional to these numbers, say
75x, etc. We then have

(20x) (101 ,450) +(75%) (1 1 1 ,360) +(200.^) (65,210)

+ (400%) (11,1 50) = 39,000, say, givdng
% 0*00140

The fractions are then approximately 2*8, 10*5, 28 and 56 per cent.

The figures used in actual practice (though not obtained by this method)
were 5, 10, 25, 50. As we shall see below, extreme precision in the
sampling proportions is unnecessary. It was recognised that the smaller
farms were over-represented, this being a deliberate modification intro-
duced for other purposes.

538

THEORY OF STATISTICS

We may form an idea of the relative efficiency of this method of sampling
as compared with others which might suggest themselves. With sampling
tractions 5, 10, 25, 50 and 100 per cent we have

l0/\

Variance
(proportional to)

(Farms

sampled)

101,450

20*

5,072

111,360

75*

11,136

65,210

200*

16,302

11,150

400*

5,575

1,430

100

—

1,430

Totals 290,600

39,515

Now from our expression for the variance of x we have

var t

We may now calculate this quantity, or rather a quantity proportional
to it (since we are assuming the variances proportional to the squares of
the widths of the grouping intervals). For instance the first term in the
summation is —

101,450 X 400 X (20-1).

We find that var t is proportional to 0-1896. We do not require the
variance of the last interval because the factor ——1 vanishes for it.

It is also of interest to see what happens if we draw the same proportion
from each of the five strata, a procedure which has a certain prior
plausibility. The total sample number is 39,515/290,600=13-598 per
cent. We shall now require an estimate of the variance in the last class
of farm of 700 acres and over, and shall take it to be proportional to 400^.
Denoting the sampling proportion by p we have, for an estimate of the
mean w based on this method,

This formula gives us var w proportional to 0-3979, i.e. a variance more
than twice as great as that obtained by the fiist method. •

23.10 From the determination of the best sampling fractions by
minimising the variance it follows that fractions near to the optimum
will give almost as good results as the best. We may establish the result
directly as follows. Let

SOME PROBLEMS OF PRACTICAL SAMPLING

539

Then

vari = .... (23.16)

Now suppose that instead of the optimum proportions we choose
proportions where the ^’s are small and may be neglected. Since
the sample number is the same in both cases we have

giving

S(iVA) = 0

If is the alternative estimate

( 23 . 17 )

var ti

— SiV

and since, to our approximation

Now is equal to where a is a constant and consequently the second
term vanishes in virtue of (23.17). Thus var u is practically the same
as var t

The effect of this result is that we need not be too meticulous in deter-
mining our sampling fractions. Any values near the optimum will give
a sampling variance very near the minimum.

23.11 Various elaborations of ordinary sampling or stratified sampling
are possible and are sometimes employed. For example, we may sample
in two stages, the second sample being a sub-sample of the members of
the first sample ; and the method may be extended to further sub-
samples. Suppose, for instance, that we require a comparatively small
sample from the inhabitants of a certain country. For administrative
reasons it may be more convenient to draw first of all a primary sample,
consisting of towns and rural districts ; then, from each member of the
sample, a number of houses ; and then, say, one member from each house.
At some stage in the process, e.g. in the selection of houses, we might
have stratified. There is evidently a very large number of possible

540

THEORY OF STATISTICS

combinations of different techniqnes in general, although in practice a
limit is often imposed by cost or convenience.

23«12 The student will inquire whether there is any advantage in these
more complicated procedures from the theoretical viewpoint ; whether,
for example, it is possible to reduce the sampling error by sub-sampling.
We shall only have the space for a brief discussion of this question.

If ail the sampling is random and the population is homogeneous there
is no theoretical advantage in sub-samphng. An ordinary random
sampling process gives each member of the population the same chance
of being chosen. If we choose groups at random, ani the members of
those groups may he regarded as having been allotted at random to the groups,
the more complicated technique also gives each member the same chance
of being chosen, and the methods are equivalent.

23.13 In practice, however, the nature of the grouping is often known
to be such that the members cannot be regarded as grouped at random,
and the effect of stratification or sub-sampHng may be to alter the
standard errors of estimation quite considerably. To take our former
example of sampling from a human population : there may be (and
usually there is) a good prior reason to expect that the quantity we are
investigating differs between town and country districts, so that the
population is patchy and, m any given area, there is a positive correlation
between contiguous members of a sample ; or again, if we take only one
member from a household we may exclude from occurrence certain
coincidences or resemblances w^hich are more likely to occur within a
household than between households. This patchiness in the population
may, or may not, be an advantage in reducing the standard error. There
do not appear to be any very general rules on the subject and a great
deal depends on the nature of the patchiness. It is nevertheless possible
to make certain assumptions about certain types of population with
great confidence, and to base sampling techniques on them.

Example 23.3. — A survey is carried out in a particular town. Certain
households are chosen at random and then one member from each house-
hold. Suppose the quantity under consideration is some continuous
variate

Let us suppose that the maximum number of members in a family is k,
that there are families with one member, F^ with two members and so
on. The total number of families we may write as F and the total number
of individuals as iV. Then we have

^F,^E (23.18)

I jF^ = N

. (23.19)

SOME PROBLEMS OF PRACTICAL SAMPLING

541

Let the mean and variance of x in the Ith family of the set of families
be m^i and respectively. Then if m and v are the mean and variance
of the total population of individuals

h Fj

Nm=l> S . (23.20)

3 I

For an unrestricted random sample of n (small compared with iV) from
the whole population the variance of the mean is v jn, say so that we
have

^ s s y{»„4-w*,)-

3 I

S 'Ljfn*i
3 I

. (23.21)

Now suppose we take a random sample of n households and choose
one member from each household. In such a case we are sampling from
a population of F members, one from each member. The variance of
such a population is given by F, say, where

V +

/I h Fi \ 2

1 fc F/

= P S S

and hence the variance of the mean of samples of n, say is given by

Vu =

S {Vji+mjt)-

i? S S
p i I

(23.22)

The reader will notice that the sampling variance can be exhibited
in the form of an analysis of variance. If v is the mean of variances
within families and is the variance of means (between families) we have

. (23.23)

From (23.21) can be put in a similar form but the mean of % is weighted
according to the number of members in a family and the sum corresponding
to the m^i is similarly weighted.

A comparison of (23,21) and (23.22) will show that if the means and
variances increase with size of family, or if the variances increase and
the means remain constant, is greater than for the larger families
then contribute relatively more to The situation might then arise
in which we had a smaller sampling variance by choosing one member
from each family in the sample. On the other hand we have to be careful
not to obtain a biased estimate. In this case, the mean of a sample of n,
one from each family, might be biased. For the mean of such a sample
(over all possible samples) is the same as the mean of one member over

542

THEORY OF STATISTICS

all possible samples consisting of one family, that is to say, is the un-
weighted mean

I Jc Fi

~ 2 S Mji .

This may differ from the population mean given by (23.20). We must
always be careful, therefore, in looking for estimates with minimum
variance, not to choose one which may be seriously biased.

23.14 At this point we may mention briefly certain other types of
sampling which are sometimes used. In some of these cases the methods
have not yet been put on a satisfactory theoretical basis and the reader
who proposes to use them should read more widely before doing so.*

(a) Systematic sampling. Where the members of a population are
arranged in some spatial or temporal order (e.g. persons listed alpha-
betically in a telephone directory, price quotations given regularly each
week, plants growing in rows in a field) it is sometimes convenient to
choose a sample by selecting members at equal intervals along the order.
For instance we may select every 100th name on a list, or every fifth plant
in a row. We referred in 16.26 to the selection of houses in a street and
the dangers of occasional bias which it might introduce. Such methods
have been called (not very aptly) systematic sampling. Where the
population is patchy they have the appearance of avoiding selecting by
chance too many members in an unrepresentative area. On the other
hand, where there are rhythms present in the population (as, for example,
in oscillatory time series or in soil which has been cultivated by machine)
the method may give very unreliable results. It can only be recommended
when there is good reason to think on prior grounds that the interval
between members of the sample has no relation to any possible systematic
properties of the population.

{h) Quota sampling . — In social surveys involving interviews when the
work has, in general, to be divided among a number of investigators it
has sometimes been the practice to assign to each a definite sample number
which he must attain — he may, for instance, be instructed to secure
200 schedules, and to go on until he has obtained that number. This
method would be unobjectionable if the sample were random, but un-
fortunately circumstances may arise which vitiate the randomness. The
investigator who meets -with refusal to complete a schedule or otherwise
fails to obtain one from a previously selected individual (e.g. because of
his absence), must go on until the quota is full, and may be forced to take
his sample where he can get it, not where he would like to get it. Checks
and controls throughout are most desirable in this type of sampling.

* See F. Yates, Sampling Methods for Censuses and Surveys, 1949, Griffin and Co., for
an extended account of the subject and a bibliography.

SOME PROBLEMS OF PRACTICAL SAMPLING

543

(c) Sequential sampling , — This method (which has been put on a
satisfactory theoretical basis, although many problems remain unsolved)
aims at economising in the size of sample required to reach a prescribed
degree of probability in making a correct decision.

In the ordinary sampling process such as we have described it in fore-
going chapters, we select a sample of pre-determined size and calculate
from it the required estimate together with its standard error (or, for
small samples, an equivalent quantity) which sets limits to the values
between which the parameter value may be stated to lie to a prescribed
degree of probability. In sequential sampling we invert the process to
some extent. We decide, on the basis of the prescribed degree of
probability, what are the limits within which we can accept the sample
estimate as consistent with prescribed parameter values and then sample
one by one. If at any stage the sample estimate (or more generally,
some suitable statistic calculable from the sample) falls outside the limits
appropriate to the size of sample which has been reached up to that point,
we reject the hypothesis that the population parameter has the prescribed
value or set of values under consideration. An excellent account of the
method will be found in A. Wald's Sequential Analysis.

Example 23.4. — As an example of an inquiry which was spoilt by
violating some of the principles we have proposed, we may take the
Lanarkshire nutritional experiment which was undertaken in 1930 at a
cost of £7,500. For four months 5,000 children received three quarters
of a pint of raw milk per day, 5,000 received the same quantity of
pasteurised milk and another 10,000 were chosen as controls. The
height and weight of the whole 20,000 were measured at the beginning
and end of the experiment.

The main object of the experiment, of course, was to see if the milk-fed
groups gained more in height and weight than the controls, but for it to
have any value as a basis of generalisation the samples had to be random.
The intentions of the planners of the experiment were good. Teachers
selected the children either by ballot or by some alphabetical system.
But at this point a serious flaw occurred. '' In any particular school where
there was any group to which these methods had given an undue proportion
of well-fed or ill-nourished children, others were substituted in order to
gain a more level selection."

It is unfair to be too critical of what was evidently a well-intentioned
procedure to improve tjie representative quality of the data ; but in
fact this attempt to balance the samples nearly ruined the experiment.
It was found at the end of the inquiry that the controls were both heavier
and taller than the fed children by about three months' growth in weight
and four months' growth in height. It appears that the substitutive
process in what looked like unusual samples resulted in the choice of
better nourished children as controls and worse nourished children as
feeders. Comparability with controls was thereby invalidated.

544

THEORY OF STATISTICS

A second object of the inquiry was to see whether there was any
differential effect between raw and pasteurised milk. Here again a
mistake was made. A particular school obtained either one kind of milk
or the other, not both. Now in a district which is racially or socially
heterogeneous, it is possible that the selection of one half of the schools
for one treatment might result in the choice of a set with higher or lower
standards than the other half, both in the original measurements and
in the rate of growth. It would have been better to select a number for
feeding with raw and an equal number for feeding with pasteurised milk
in each school.

There were other faults in the design of the experiment and the majority
of the conclusions which were drawn from it did not, strictly speaking,
follow from the data. The student may consult Student/' Biometrika,
1931, 23, 398 for some further criticisms.

Examination of samples

23.15 The liability to error of the result of examination of a sample
unit obviously depends to a high degree on the nature of the observation
to be made. A simple physical measurement permits of a high degree
of accuracy with little chance of bias, but even here care must
be exercised, e.g. in taking body-measurements on the human subject
to determine correctly the points between which the measurement is
to be taken, and to use a constant degree of pressure in adjusting the
instrument. If an estimate is made, the possibility, indeed the probability,
of error is at once greatly increased, as we have seen already in the estima-
tion of shoot-height (16.21). The chances of error are widened yet further
stiU if the unit is a human being and makes his own contribution towards
misleading the observer, by giving untrue or ambiguous answers to his
questions. In such interviewing work a knowledge of and familiarity
with psychology may be of far more service to the investigator than a
knowledge of statistical method. We will give some examples first of
estimation and secondly of interviewing that will serve to illustrate the
risks.

Example 23.5 . — Corrections for pessimism

Table 23.1 shows the forecasts of yields in potatoes made on various
dates as compared with final estimates, for a series of years.

These forecasts and estimates are averages based on figures supplied by
a number of estimators scattered over England and Wales. They are not
checked against actual yields, although some estimators use known results
in their areas for particular farms and fields in arriving at their judgment.
The striking thing about the figures is the uniform sign of the difference
between the forecasts and the final estimate.

This type of bias is quite different from the one noticed in the Example
23,1. There the investigators measured the yield of definite areas and the
bias apparently lay in their enthusiasm in extending those areas a little

SOME PROBLEMS OF PRACTICAL SAMPLING 545

TABLE 23.1, — Forecasts of yields of potatoes in England and Wales in tons per acre

From the official agncultural statistics

Year

Sej

)t. 1st

Oct. 1st

Nov. 1st

Final

estimate

Yield

difference
from final

Yield

difference
from final

Yield

difference
from final

1929

5-7

-17-4

6-2

-10-1

6*5

- 5*8

6-9

1930

6*0

- 7-7

6*1

-- 6-2

6*1

- 6-2

6-5

1931

5-5

0-0

5-3

~ 3*6

5*3

- 3-6

5*5

1932

6-4

- 3«0

6-2

- 6*1

6*3

- 4«5

6*6

1933

6*4 1

- 4-5

6-2

- 7-5

6«4

- 4-5

6-7

1934

6-0

-15*5

6-3

-11-3

6*7

- 5-6

7-1

1935

5*6

- 9-7

5-7

- 5-1

6*0

- 3-2

6-2

1936

6-0

- 3-2

5*9

- 4-8

5*8

- 6-5

6-2

too widely. Here the investigators are not measuring but judging and
the bias arises from excessive caution, a kind of chronic pessimism which
is well recognised in agricultural circles. The remedy would be either
to lay down a series of harvesting experiments on properly chosen sites,
or to correct forecasts in future by scaling them up proportionately
to the average deficiency over a previous series of years. Given time,
of course, it might also be possible to educate the observers out of their
pessimism, but this would not be without its dangers and might for a
time swing the balance the wrong way.

Example 23.6, — When an investigator is sent out into the field to collect
results he may, if he is lazy or dishonest, shirk his duties and send in
returns which are spurious. Once these faked records have occurred it
is difficult to detect them unless the inquiry has been specially designed
to be self-checking in this respect, but various methods are available to
check the general accuracy of the individual or to restrain his tendency
to make entries by guesswork. One useful device is to have a second
investigator cover some of the same ground. This results in a certain
amount of duplication of effort but is often worth the extra trouble and
expense. The two investigators need only have part of their field in
common. The knowledge that any particular return is likely to be
checked by another investigator is often a sufficient spur to accurate
recording in all the records.

Table 23.2 shows a comparison of two recordings by surveyors A and
B made on identical fields within a fortnight of each other. The surveyors
merely had to record the crop xmder which each of 332 fields lay and no

546

THEORY OF STATISTICS

question of measurement or estimation was involved apart from the
identification of the plants.

TABLE 23.2. — Comparison of duplicated complete enumeration in a district of Bengal

(Mahalanobis, J. Roy, Slot. Soc , 1946, 109, 325.)

A — Survey

Jute

Winter

nee

B — Survey

Winter nee
and jute

No. crop

Totals

Jute.

Monsoon nee

Monsoon nee and jute

Jute, monsoon and
winter nee

—

Riee (monsoon and
vnuter)

—

No crop .

102

188

Totals

140

118

332

The discrepancies are obviously very large and it is impossible to avoid
the conclusion that one of the surveyors at least was not carrying out
his duties properly. Errors on this scale can hardly be due to accident
or inability. There is a strong presumption that one of the surveyors
at least was either not exercising reasonable care or definitely falsifying
his records,

23.16 Unintentional errors on the part of investigators can to some extent
be eliminated by training and careful instruction, and the magnitude of
unconscious bias can often be gauged by letting them undertake a dummy
inquiry on material for which the results are known. Where resources
permit, however, it is very valuable to replicate the inquiry among
different observers to see how far they differ among themselves. This
is especially desirable in inquiries which necessarily depend on subjective
judgment, such as the assessment of a candidates' qualities in a personal
interview, a grading by an inspector of the suitability of a house for
habitation or the rating of an employee for promotion.

Example 23.7. — In an inquiry into family budgets in Nagpur
(Mahalanobis, /. Roy. Stat. Soc., 1946, 109, 325) information was collected,
inter alia, of total income and of monthly expenditure. The area under
examination was divided into five zones. Within each zone samples were
selected by picking families at random and these were divided into four
sub-samples, each of which was random and independent of the others.
There were four investigators, each taking one sub-sample at random

SOME PROBLEMS OP PRACTICAL SAMPLING

547

in each zone. Within each sub-sample about 50 schedules were collected.
Thus the total of about 1,000 schedules (actually 997 because of small
imperfections in carrying out the design) can be classified into a 5 X 4 X 50
grouping, and the variance-analysis is of the following form.

TABLE 23.3. — Nagpur Family Budget Inquiry

Analysis of Variance
(For ref see Table 23 2)

Variation

d.f

Quotient

(Income)

Quotient

(Monthly expenditure)

Between zones (Z) .

4,439 6

3,707-9

Between investigators (I)

85-4

597-1

Interaction (ZI)

382-5

397-3

Between sub-samples

1,189-7

1,127 1

Within sub-samples .

977

401-6

384 7

Total

996

424-7

398-9

We have shown only the degrees of freedom and the quotients in the
table. If the reader multiplies the two to obtain the sum of squares
he will find that the sums '' between sub-samples and " within sub-
samples do not add to the total sum. This, of course, is due to the
fact that the numbers in sub-classes are not units but are about 50.

The analysis shows the interaction between zones and investigators.
If there were only one schedule in the sub-sample there would only be
19 degrees of freedom altogether ; but as there are about 50 schedules in
the sub-samples we can form an estimate of the variance within sub-
samples by taking the variance of each set of schedules in a sub-sample
and pooling for the 20 sub-samples. It is this residual variance
(401*6 for income and 384*7 for monthly expenditure) which is to be
compared with the other variances to test departure from homogeneity.

Taking income first, we find that the ratios of the residual quotient
to the quotients between investigators and the interaction are not
significant. This is encouraging and indicates that the investigators are
accurate (or at least consistent). The quotients between zones and
between sub-samples are significant at a 1 per cent level. This was to
be expected from the nature of the inquiry, for the zones were deliberately
chosen from differentiated areas.

A similar conclusion is reached in respect of monthly expenditure.
The reader can verify the arithmetic of the significance for himself.

23.17 To avoid confusion we refer at this point to a technical meaning
of the word “ bias " which has recently come into use in .advanced
theoretical statistics. A statistic t which is used as an estimator of a

TABLE 23.4. — Bengal crop survey

Compansoa of two independent estimates of proportion p under winter nee

548

THEORY OF STATISTICS

SOME PROBLEMS OF PRACTICAL SAMPLING

549

parameter 6 is said to be biased if the mean value of i over all possible
samples is not equal to 0, Thus, as we saw in 21.4, the sample-variance
is a biased estimator of the parent variance because the average value of
over all samples is (n—l) /w times instead of fi^ itself. To obtain an
unbiased estimator we must use the statistic

s'2

The meaning attached to the word " bias in this chapter is not
restricted to departure from the criterion we have just mentioned. In
the narrower sense of that criterion bias is a quahty of the estimator
employed and may exist when the sampling is random. In the more
general sense bias may be used to connote any effect which distorts the
representativeness of the result, whether in the estimating process or in
the selection and examination of the sample.

Cumulative effect of bias

23.18 There is a popular belief that even if individuals make mistakes
their errors in the aggregate will tend to cancel out, so that an average
of a number of instances will be less distorted by bias than any particular
single instance. To some extent this is true. If the errors are in the
nature of sampling fluctuations we know that the standard error of a mean
decreases proportionately to the square root of the number of observations.
But it would be a mistake to assume that all types of bias tend to be of
the self-cancelling kind. It is not true that if only enough people make
enough mistakes the average of their opinions or estimates lies near the
real value,

23.19 We have had one example of the cumulative effect of bias in
Example 23.5, in which we saw that, in spite of the number of crop
estimators concerned, the mean of their forecasts was systematically
below the final estimate. Evidently they were all affected more or less
by the same tendency which therefore persists in the average of the
individual results. How far, in any particular inquiry, we may assume
that individual biases tend to cancel in the aggregate depends on the
nature of the inquiry. We clearly cannot assume that there is safety in
numbers where individuals may be affected by the same kind of bias,
e.g. if there is any general tendency to over-estimate for reasons of personal
pride, or where some force is at work to remove from the sample individuals
of one particular type. On the other hand, cases are known wherein
biases (not merely chance fluctuations) do appear to cancel themselves
out very largely.

Example 23.8. — (Data from Mahalanobis, loc. cit Example 23.7).

A certain area of 6,204 “ grids '* of about 2J acres each was surveyed
independently by two parties A and B. Each party recorded for each
grid the estimated proportion under winter rice. The results are shown
in Table 23.4.

550

THEORY OF STATISTICS

If the two parties were in complete agreement only diagonal cells
would contain non-zero entries. The differences are evidently quite
substantial, there being only 51-6 per cent of the cases showing complete
agreement.

Nevertheless the mean of p for A (mean of column totals) is 52*0 per
cent, whereas that for B (row totals) is 51*9 per cent, an extraordinarily
close agreement. Thus, in spite of the differences on individual grids the
estimates for the whole are satisfactorily concordant.

Example 23.9. — The vanity “ effect.

The preceding examples have related to defects on the part of the
observers. We now consider a different type in which bias is introduced
by a distorted response from the members of the samples.

In an inquiry into listeners’ preferences for radio programmes subjects
were asked by interview for their opinion on broadcast religious services.
52 per cent of the persons indicated by their response, in the interviewer’s
judgment, that they were enthusiastic or moderately enthusiastic. One
might have been tempted to infer that about half the listening public were
keen listeners to religious broadcasts. In fact the listening audience
seemed to be about 10 per cent of the listening population, another and
more direct inquiry into the audition of actual programmes giving
proportions ranging from 3 per cent to 18 per cent (See Silvey, J, Roy,
Stat, Soc,, 1944, 107, 190 for details).

Without dwelling on questions of standard error we can see at once
that the responses in the interviews were strongly biased. There can be
little doubt that this was due to the wish on the subject’s part not to
be classified as indifferent to spiritual influences. The same kind of effect
is apt to arise in any inquiry into cultural tastes, few people being willing
to admit to a stranger that they do not care for good music, however
rarely they go to the trouble of listening to it.

Example 23.10. — The “ sympathy ” effect.

The Listener is a British weekly journal devoted to broadcasting matters.
An inquiry was made to find out how many people read it. Now in this
case the circulation of the journal is known and, by making due allowances
for the numbers of people who read the same copy in family units, a fair
estimate can be obtained of the total number of people who can possibly
read one issue. The percentages obtained from sampling inquiries showed
that four or five times as many people said they read it as could have done
so. (See the remarks by Durant on the paper by Silvey referred to in
the previous example.)

It would not be correct to deduce that the majority of the people
replying affirmatively to the question whether they read the Listener are
deliberate liars. There is a natural tendency on the part of many people
to give to the questioner the reply which they think would please him.
They infer that an affirmative answer would do so (thinking, perhaps,

SOME PROBLEMS OF PRACTICAL SAMPLING

551

that the questioner is a representative of the publishers) and stretch their
consciences to the extent of saying that they read the journal when they
may, for instance, only have seen it on a bookstall or in a friend's house,
or even if they have merely seen it advertised. This sympathy
response is all the more difficult to guard against because the interviewer
must try to ingratiate himself with his subject in order to obtain a reply
at all.

In this particular case there is another possible explanation of the bias.
The subject may imagine that if he gives a negative response an attempt
will be made to sell him the journal. He therefore anticipates any possible
sales pressure by stating that he takes the journal already.

23.20 The lessons to be learnt from such experiences as these are
numerous. We will indicate a few methods which the investigator
may sometimes be able to use to minimise the risk of the distorted
response.

(a) If possible the aim of the inquiry should be concealed from the
subject. This will prevent him from “ co-operating " with the interviewer
to get what he may consider the desired result. But it is often im-
practicable to expect him to answer questions without asking some in
return ; and very often the purpose of the inquiry is clear merely from the
fact that it is made.

(b) The questions should be framed unambiguously so as to elicit a

yes-no " response or a three-way answer customary in opinion inquiries :

yes /no /don't-know

(c) Independent checks on veracity can sometimes be obtained in a
roundabout way. In Example 23.9 we mentioned a case where a direct
check was available. An inquiry on a political subject, for example, may
well embody some question which permits of checking against known
results for the aggregate, such as '' Did you vote at the last election ? ”
The interpretation of the results of these “ control " questions is not
always very easy, but they provide valuable collateral evidence on the
general representative character of the responses.

(d) If there is prior reason to suppose that different types of subject
will give varying degrees of distortion in response, results for the types may
be analysed separately. Suppose we are conducting by personal interview
an inquiry which involves recording the subject's age. Knowing that the
incentive to lie about age varies from one age-group to another, we may
analyse the replies, if they are sufficiently numerous, into age-groups.
From known census data or by making certain assumptions about the
population under examination based on known facts such as birth-rates
and death-rates, we can estimate what the results ought to be if the
subjects are telling the truth, and hence gauge the direction and extent
of the bias.

THEORY OF STATISTICS

SUMMARY

1. The complete sampling process consists of (a) the choice of unit,
(b) the selection of the sample of units and (c) the examination of the units.

2. For '' continuous regions there is usually no natural unit ; and for
a disconcontmuous population practical considerations may suggest, as
size of unit, groups of the individuals comprising the population.

3. By the use of appropriate variable sampling fractions in stratified
sampling a considerable reduction may be made in the sampling variance
of estimates of the mean. For linear estimates the optimum estimate is
given when the numbers taken from the strata are proportional to the
standard deviations of the variate under investigation in those strata.

4. Various examples are given of the introduction of bias, due to flaws
in the '' examination of the sample.

EXERCISES

23. 1 Consider possible sources of bias in replies to the following enquiries :

(a) Persons are asked to state how often they attended a place of
entertainment during the previous year ;

(b) Persons are asked to state how many days have elapsed since they
last attended a place of entertainment.

Consider how far the answers to (6) may be used as a check on the answers
to (a).

23.2 Ten investigators are to be sent to ten trafiflc centres in a city to
record the number of automobiles passing a specified point in a specified
time. Two of the investigators are suspected of being unreliable. Design
a method of carrying out the inquiry which will exhibit this unreliability,
if it exists, and will also provide unbiased results if the other investigators
are reliable.

23.3 A number of businesses are asked to provide figures showing stocks
of specified goods on hand at a specified date, and the returns are required
within a specified and rather short time. Consider what kinds of bias
might appear in the answers.

23.4 A random sample is drawn from the records of a fire insurance
company with the object of estimating the number of fire '' incidents
occurring in a certain period in dwelling houses. Consider how far this
sample is likely to be unrepresentative of all fire incidents '' which require
the attention of a public fire service.

SOME PROBLEMS OF PRACTICAL SAMPLING

553

23.5 If equation (23.10) may be accepted as self-evident, provide a
simplified proof of the result of equation (23.11). In the manner of 23.10
derive equation (23.13).

23.6 A population is stratified into four (large) groups for which the
number of members and the variances are as follows —

Group

Number

Variance

10,000

20,000

40,000

30,000

Find the variance of an estimate of the parent- mean based on a sample
of 400 from the population

{a) by taking 100 from each stratum ;

(6) by taking a constant proportion 0 • 4 per cent from each stratum ;
(c) by choosing the sample numbers (to the nearest unit) proportionately
to the standard deviations in the strata ;

{d) by taking the sample numbers as the optimum, as given by (23.15).

23.7 A population consists of N members in order, divided into k groups
of n. A sample is selected by taking the yth member of each group, so
that it is systematic and consists of the members
Show that the variance of the mean of the sample, say x, is given by

var ^ = I

l+[k-\)p

where v is the variance of the population and p is the intraclass correlation
coefficient of the n groups of k consisting of 7th members (^=1, - . . h).
Hence show that var x is greater than, equal to, or less than the variance
of a random sample according as the intraclass correlation is positive,
zero or negative. It may be assumed that N is large compared with k.

23.8 A sample is drawn from an ordered population of N{—kn) members
by dividing it into k sets of n and taking a member at random in each of
the k sets. Consider generally whether the variance of the mean of such
a sample will have a smaller variance than the mean of an unrestricted
random sample.

23.9 One of the main difficulties in house-to-house inquiries is to make
proper allowance for those houses where there is no one at home when the
call is made. It has been suggested that suitable methods of dealing
With this problem would be

THEORY OF STATISTICS

(a) to call back persistently until an occupant was found to be at home ;

(b) to sub-sample the non-responsive houses by calling back persistently
at a proportion of them ;

(c) if possible, to stratify houses beforehand according to the proportion
of the day during which somebody was at home, and to sample at
random in each stratum, ignoring the non-responders.

Examine the relative merits of these methods.

23.10 Discuss the problems of obtaining estimates of average annual
values in the following cases *

{a) Expenditure of persons on holidays by sampling at various dates
in the year ;

(b) rainfall at a certain locality by sampling for rainfall on a specified
number of days ;

CHAPTER TWENTY-FOUR

INTERPOLATION AND GRADUATION

Simple interpolation

24.1 If the value of a function of a single variable say has been
tabulated for equidistant values of the variable Xy x-\-hy x-{-2h, etc., we
often require to find the value of the function corresponding to an inter-
mediate value of the variable. Functions in very general use, such
as common logarithms, have usually been tabulated with intervals so small
that even over a range of several intervals the relation between Ug. and x
may be assumed to be effectively linear, that is of the form

Ug. = aQ~\-a^x .... (24.1)

as is shown by the constancy of the differences between successive values
of w. For example,

TABLE 24.1

Number

Logarithm

Difference {4-)

30v597

4-4856788

0-0000142

30598

4-4856930

0-0000142

30599

4-4857072

0-0000142

30600

4-4857214

0-0000142

30601

1 4-4857356

0-0000142

30602

> 4-4857498

If we then require, say, the value of log 30600-3, it is sufficient to use the
familiar process of simple interpolation —

log 30600 4-4857214

0-3x0-0000142 43

4-4857257

The little multiplication sum, is, in most tables, already done for us in the
margin.

555

THEORY OF STATISTICS

05 ^

Differences

24.2 For any function which has been tabulated to sufficiently fine
intervals (within certain limitations) simple interpolation can be used in
this way — it is only a question of making the intervals sufficiently small
(see below, 24.16). But many functions have not been tabulated in such
detail, successive differences are not equal, and consequently simple
interpolation cannot give an accurate result. The problem then arises,
how are we to interpolate with reasonable precision ? And the answer is
given hy proceeding to higher orders of differences, as they are termed ; i.e.
instead of considering only the differences

Ao^ =

Aa^ = ^2

etc,, we also consider the second differences

Ao'-Ai^-Ao^

A2^ = As^ A2^

etc., or even the third differences, fourth differences, etc.

24.3 To take an actual example, Table 24.2 shows the squares of the
first few natural numbers, together with their first and second differences.
Following a practice which is convenient for printing and for most purposes
of practical work, each difference is printed, not on a hne between the
two figures to which it relates, as with the logarithms in Table 24.1 above,
but on the same line as the upper figure of the two concerned — the line
of the figure subtracted ; and as the signs of the differences are constant
for each column this sign is simply stated at the top.

TABLE 24.2

Number

Square

First diff

AH+)

Second difi.

AH + )

Third diff

—

— ,

—

— —

—

Here we see that the first differences — the only ones with which we
have been concerned hitherto — are no longer constant ; but they follow a
simple rule, in that they are an arithmetic series, a linear function of x.
As a result, the second differences are constant, actually +2,^ and con-
sequently the third diSerences vanish.

tNtERPOLATlON AND GRADUATION

557

24.4 The figures on the first line of such a table are called the leadmg
term (0) and the leading differences (+1, +2, 0), and it is evident that,
given the leading term and the leading differences, the whole table could
be built up by successive addition as far as we pleased, without calculating
any square directly except for checking. The series of first differences
would be obtained by adding 2 over and over again, starting from the
leading difference 1, i.e. 1+2=3, 3+2=5, etc. The squares would be
given then by adding these differences in succession to the leading term 0 :
0+1=1; 1+3=4; 4+5=9, etc.

Differences of a polynomial

24.5 From these results we may conclude quite generally that the second
differences of any polynomial of the second degree,

Uy,— aQ-{-a^x+a^x^ .... (24.2)

are constant and the third differences vanish. For, if we multiply all the
squares in Table 24.2 by any factor ag, we merely multiply all the differences
of every order by the same factor ; and the linear part of the function,
ao+^i^, cannot contribute to second differences.

Below we give a similar table, Table 24.3, for the cubes of the first few
natural numbers, and here it will be seen that third differences are constant

TABLE 24..3

Number

Cube

First difi.
A‘(+)

Second diff

A*(+)

Third difi
A»(+)

Fourth diff
A*

—

125

—

“““

and fourth differences vanish. By similar reasoning we may conclude
that the third differences of any polynomial of the third degree,

• * • (24.3)

are constant and the fourth differences vanish. The student will be quite
correct if he draws the general conclusion that for a polynomial of the fth

degree, _ ^^o+^^^A;+^^ 2 ^^+ . . . . . (24.4)

the rth differences are constant and the (f+l)th differences vanish. To
prove this it is only necessary to note that each successive differencing
lowers the degree of a polynomial by unity, for the difference of any term
is

... 4-1

X , /i

which is a polynomial of degree (A~l).

558

TlJ£ORY Of StAtlSTiCS

Newton’s formula

24.6 Evidently these results hold out some possibility of generalising
our method of interpolation. If, instead of only considering two successive
values of say and and using the linear relation between and x
that will reproduce these values to give any required intermediate value
of we can use the polynomial of the second degree which will reproduce
three adjacent values, u^, or that of the third degree which will

reproduce four, Uq, and evidently we shall be likely to get much

more precise results. But to do this we must be able to obtain the required
polynomials in terms of the differences. We shall use the notation already
introduced, i.e.

Function

First diffs.

Second diffs

Third diffs.

Fourth diffs

Ao"

Ao*

Ai»

Ai"

Aj*

—

Further, the common interval for the values of x will be taken as unity,
as shown ; in practical work this is always treated as the unit until the
end of the work, just as the class-interval is so treated when calculating
the moments of a frequency-distribution.

24.7 Now write down the leading term and leading differences at the
head of a table with spacious columns, as below, up to the leading fourth
difference, and fill in the rest of the table working back from right to
left. In column 5 for third differences we can fill in only the second
space, Ao^-f-Ao^. In column 4 for second differences the second term
will be Ao^-fAo^ (always adding from the line above to the right) ; the
third term will be Ao2-f.2Ao^+Ao^. We leave the student to supply the
remainder.

Third

Fourth

First diffs

Second diffs

diffs

A„*

A »

Ao‘

Wi:=Wo-{-Aol

Ao^+Ao^

Ao^+Ao®

AoHAo*

—

%=?=Wo-|-2AoHAo2

i Aoi+2Ao2H-Ao«

‘ Ao2 + 2Ao"4-Ao^

—

^0 4“ 3Ao^ 4" ^Aq^ -f Aq®

AoH3Ao2+3Ao®+Ao^

—

t*4-*^o4-4AoH6AoH4Ao®4-Ao^

—

INTERPOLATION AND GRADUATION

559

Now look at the numerical coefficients in the expressions for Uq, u^,
etc. ; they run

1+1
1+2+1
1+3+3+1
1+4 +6+4+1

These are familiar figures ; they are the terms in the binomial expansions
of (1+1)®, (l+l)^ (1+1)^ (1+1)®, etc. We then have, generally,

(24.5)

where the series of differences may be continued so far as is necessary to
give a result of the precision desired. This important equation is known
as Newton's Rule or Newton's Formula. It may be repeated that in this
form of the equation the unit of ^ is the interval. There are many other
formulae of interpolation, but we propose to limit ourselves to this and
illustrate its uses.

24.8 It will be seen that, if the series on the right of (24.5) is terminated
at Aq', the expression is a polynomial of the rth degree in x, though it
is not arranged according to powers of x but according to the successive
orders of difference, which is more convenient for our present purpose.
This polynomial passes through the r+1 successive points (0, (1, %),

(2, . . . (r, u^. In particular, if the series terminates at Ao^ we

have simple interpolation and the polynomial reduces to the straight line
passing through (0, and (1, %). If it terminates at Aq^, the series
represents a parabola of the second degree passing through the three points
(^> %)> (2, u^. If it terminates at Aq®, it represents a polynomial

of the third degree passing through the four points (0, (1, (2,

(3, ; and so on. But the student must remember that even though

the polynomial reproduces the values of the function at 0, 1,2 and 3, it
does not necessarily closely reproduce the function at intermediate values
of X. The whole utility of the formula is dependent on the closeness with
which the variable can be represented locally by a polynomial of fairly low
degree. Most ordinary functions satisfy this condition when tabulated
for small intervals, but occasionally the student may find himself in
difficulties. We will give some examples in later sections.

We now proceed to some illustrations, and will give a warning at once :
the student must he very careful as to signs.

Example 24.1. — Given the cubes below, required to find the cube of
32*4.

We give this first as an example in which the interpolation is exact,
for the third differences are constant, so that we need not proceed further,

56 o

THEORY OF STATISTICS

Number

Cube

AM+)

A®(+)

29791

2977

192

32768

3169

198

35937

3367

204

—

39304

3571

—

42875

—

As interpolation is exact, it does not matter which term we take as
Uq, Supposing we take an origin at a; =32, Then for 32-4, and

we have —

^^0*4 y ~2 r~2~3

= 32768 +0 * 4 ( 3169 ) -0 • 12 ( 198 ) +0 • 064 ( 6 )

= 32768 + 1267 - 6 - 23 - 76 + 0-384

= 34012*224

This may be verified by direct multiplication, or from’ Barlow's Tables:
the student is recommended to carry out a check by taking an origin at
;^=31

Example 24.2. — Given the following cube roots, find the cube root of
102*5. The differences have been written, as is frequently done, without
the insertion of the decimal point.

Number

Cube Root

Ah+)

A^(-)

A5( + )

101

4-6570095

153192

997

102

4-6723287

152195

983

—

103

4-6875482

151212

—

104

4-7026694

—

Here, if we wish to attain the greatest possible precision and include the
third difference, we can only take an origin at 101; :3c: is then 1*5, and

W 1.5 = Wo+1-5AoHO-375Ao2--0*0625Ao®

= 4 • 6570095 +0 • 02297880 -0 * 00003739 **-0 * 00000009
= 4*67995082

Here we have retained an extra place of decimals throughout the arith-
metic in order to get the seventh place correct in the final result, and must
round this off to 4.6799508. Even so, we cannot avoid the effect of errors
in our data, viz. the errors of rounding off, in the seventh place of decimals,
the tabulated cube roots : the seventh place in our answer is still liable
to an error of ±1 to ±2 for this reason.

It may be noted that, as differences converge so rapidly in this example,
simple interpolation would give an error of little more than a unit in the
fifth place of decimals.

INTERPOLATION ANt) GRADUATION 561

Example 24.3. — From the table of Ordinates of the Normal Curve
(Appendix Table 1) find the value of the ordinate at a; /o-=0*045.

We give this example partly as a warning to the student to see that
his differences are converging so as to be likely to give a good result.
The second difference is numerically much larger than the first, viz.
392 against 199 ; he must then look at the third as well ; if this’be large
also, he may have to go to a high order of differences to get precision.
But the third difference is only +18 and the fourth diference smaller
still, so third differences will suffice for the highest precision attainable
with the five-figure table. Note that the first difference is negative, the
second negative, the third positive, and since the interval is 0*1, ^==0'45,
not 0-045.

In the difference terms we have retained two decimals beyond the
five during the work (separated by a comma) —

= '^o+0-45Aoi-0-12375Ao2+0-0639375Ao^

= 0-39894 -0-00089,55 +0-00048,51 +0-00001,15

= 0 - 39854 rounded off to the fifth place

Interpolating in the seven-figure table. Table II in Tables for StatisHctans
and Biometncians, this is found correct to the last place. It may be
noted that, if a calculating machine is used, the products given by succes-
sive terms can be cumulated on the machine.

Interpolation of statistical series •

24.9 So far we have dealt with straightforward interpolation of tabulated
mathematical functions. But interpolation may also be employed on
statistical series, or series of figures founded on statistics, provided at
least that they run tolerably smoothly. No statistical series or series
founded on statistics does, however, run absolutely smoothly, like a
mathematical function, unless of course it has been deliberately

graduated ” to do so. It must be recognised, therefore, in such cases
that we are merely using interpolation as a method of estimating the truth ;
and the truth in all probability would not and could not be given by any
process of interpolation.

The following is an illustration of a series based on statistics.

Example 24.4. — In Part II of the Supplement to the 75th Report
of the Registrar-General for England and Wales, abridged life-tables
were given for a number of counties, etc. The table below shows the
expectation of life at ages 25, 35, etc. to 85, based on the mortality of
males in Cambridgeshire in 1910-12, i.e. the average number of years
that individuals would have lived from the given age onwards, if subj ected
at each age to the mortality mentioned. Required, to interpolate values
for the expectation of life at ages 30, 40, etc.

562

THEORY OF STATISTICS

Age

Expectation
of life
(Males)

42-21

- 824

+ 20

+ 34

33-97

- 804

+ 54

+ 27

25 93

- 750

+ 81

+ 76

18-43

- 669

-f 157

- 3

11-74

- 512

+ 154

—

6-62

- 358

—

. . _

3-04

—

Total

—

-3917

+ 466

+ 134

Bottom figures less top

-39-17

-f“ 466

+ 134

—

Tables of mathematical functions will often give the differences, but
in dealing with data of this kind the student will certainly have to form
them himself, and should carry out the check shown. Having formed the
column of first differences, he should take the total, of course paying
attention to signs. In this case the total of first differences is —3917,
or inserting the decimal point, —39 'I?. This obviously must be equal
to the difference between the bottom figure and the top figure in the
preceding column, as we see is the case. The following columns must
be checked similarly.

The second differences are considerably smaller than the first differences.
Third differences are also small, but rather irregular ; it will be found,
however, that the contributions of the third differences affect only the second
place of decimals in the function, so we ought to attain a very fair result.

To get the figures for ages SO and 40 we have not much choice and must
use the known values at ages 25 to 55. On general grounds it seems
best to keep the value of x for which we require % near the centre of the
values used for interpolation. So the expectation at 50 was determined
from the values at 35 to 65, that at 60 from the values at 45 to 75, and
that at 70 from the values at 55 to 85. The expectation at 80 was
determined with the use of the second difference only from the values at
65, 75, 85.

The work is quite straightforward and the results were: 30, 38*09;
40, 29*90 ; 50,22*10; 60,14*94; 70,8*99; 80,4*64. The student
may find it instructive to draw a chart.

But some qualms were felt as to how far the results could be trusted.
A pol 3 momial is not a very good function to represent an empirical function
of the present kind which is slowly dropping to zero (see below, 24.12).
It might possibly be more appropriate to take logarithms of the expect-
ations, interpolate between the logarithms and then convert back into
numbers. The test was carried out as a control. The following are then
the data and the differences —

INTERPOLATION AND GRADUATION

563

Age

log

(Expectation)

1-62542

-0-09432

-0-02298

-0-00799

1-53110

-0-11730

-0-03097

-0-01662

-1-41380

-0-14827

-0-04759

-0-00536

1-26553

-0-19586

-0-05295

-0-03623

1-06967

-0-24881

-0-08918

—

0 82086

-0 33799

—

0-48287

—

Total

-1-14255

-0-24367

-0 06620

Bottom figures less top

-1-14255

-0-24367

-0-06620

—

The work was done exactly as before, except that the expectation at
80 was obtained with three differences from the given values at 55 to 85.
The results differed only very slightly from those obtained before, the
following table giving a complete comparison —

Age

Interpc

)lation

Difference

Direct

Loganthmic

42-21

38-09

38-07

-0-02

33-97

—

29-90

29-91

+0-01

25-93

—

22-10

22 11

+0-01

18-43

—

14-94

14-92

—0-02

11-74

—

8-99

9 00

+0-01

6-62

—

4-64

4-63

-0-01

3-04

The differences are almost immaterial.

Notes on the practical work

24.10 Number of differences to use , — Provided differences converge fairly
rapidly and continuously, there is little difficulty in coming to a decision.
The student knows to how many digits he desires to be accurate, and it
is no use his going on to higher orders of difference which affect only
places beyond this ; if he wants four-figure accuracy, it is no good his
going on to differences which affect only the sixth and seventh places.
To enable him to see more quickly the approximate contribution that
a difference of any order will give, the following table of the binomial
coefficients may be useful —

564

THEORY OF STATISTICS

TABLE 24.4. — Table of the binomial coefficients in Newton’s formula from
to x^2 by intervals of 0*1

x{x~\]

x{x-l){x-2)

x{x~l){x-2){x-3)

1 2

1 2 3

12 3 4

0*1

-0 045

+ 0 0285

-0 0206625

0 2

-0*08

+0-048

-0-0336

0 3

-0-105

+ 0-0595

-0-0401625

0*4

-0-12

+ 0 064

-0 0416

0*5

-0 125

+ 0 0625

-0-0390625

0 6

-0-12

+ 0-056

-0-0336

0-7

-0 105

+ 0 0455

-0-0261625

0-8

-0-08

+ 0 032

-0-0176

0*9

-0 045

+ 0-0165

-0-0086625

1«0

4-0 055

-0 0165

+ 0-0078375

1-2

+ 0 12

-0 032

+ 0-0144

1*3

+ 0-195

-0 0455

+ 0-0193375

1*4

+0-28

-0-056

+ 0-0224

1*5

+0-375

-0 0625

+ 0-0234375

1-6

+0-48

-0-064

+ 0 0224

1-7

+ 0-595

i -0-0595

+ 0-0193375

1 8

+ 0 72

-0-048

+ 0 0144

1 9

+ 0 855

-0-0285

+ 0-0078375

2-0

+ 1

1 0

A word of warning may, however, be desirable. Because the use of the
(f+l)th difference would not affect the result in the Ath figure, it does
not necessarily follow that this polynomial value will agree with the true
value of the function to the ^th figure.

If differences do not converge rapidly and continuously, this is in
itself evidence that a pol5momial of moderately high order does not fit
the function well and high precision cannot be expected. The student
may occasionally find himself faced by cases more difficult than those of
the foregoing illustrations. For example, here are the initial values of
P for values of proceeding by unity, and degrees of freedom
from Table XII in Tables for Statisticians, etc., Part I —

1 -000000

0-543813

0-985612

0-423190

0-919699

0-320847

0-808847

0-238103

0-676676

0-173578

INTERPOLATION AND GRADUATION 565

If we wish to find by interpolation the value at, say, 0*5, apparently we
have no choice but to take our at zero, for the table starts there. If
the student begins work accordingly, he will find his diferences not
behaving at all nicely , the second leading difference is much greater than
the first , the third is a good deal less, but the fourth, fifth and sixth
much larger than the third, and it is not until the seventh and higher
differences that definite convergence seems to be setting in. If he
laboriously works step by step, getting successive approximations to the
value of P at 0 • 5 by using one difference, two differences and so on, he
will get a series of very slowly converging values —

1. 0*992806

2. 0*999247

3. 0*999658

4. 0*998993

5. 0*998445

6. 0*998131

‘7. 0*997973

8. 0*997899

9. 0*997865

The true value is 0*997839, and he could have obtained this much quicker
by direct calculation ; even with the nine differences he has got only four-
figure accuracy. But he ought not to have expected a good result if he
had taken the trouble to look at the run of the diferences. The figures
give another useful warning. Using three diferences, we have a worse
result than when using two only. Increasing the number of differences by
one step does not necessanly increase precision.

Limitation of the number of diferences suitable for use, owing to the
efect on diferences of errors of rounding of, is considered below (24.14
and 24.15).

24.11 Choice of the set of us . — To interpolate, say, at :3f=2*5, using
third diferences, one might employ either the u’s at 0, 1, 2, 3, or those
at 1, 2, 3, 4, or those at 2, 3, 4, 5 ; one would not go outside these limits or
one would have to extrapolate for the value at 2*5, and that would obviously
be unsafe. Which set is it best to choose ? Advice cannot be absolutely
defiimte, but it would seem that usually (but not necessarily) values about
equidistant from that sought should be equally valuable as guides, and on
this principle we should try and keep the value sought so far as possible
central to the set of ^'s employed.

This suggests that one reason for our getting so poor a result above was
that we used such a lop-sided set of u's, with the value sought apparently
unavoidably near one end. Let us avoid this by a device. Repeat the
value of P for -fl at — 1 on the other side of zero. (It is true that this has
no physical meaning, but the function might conceivably run symmetric-
ally on either side of zero, and its graph has clearly high-order contact with

566

THEORY OF STATISTICS

a horizontal tangent at zero.) Now take the four values at —1,0 +1, +2
and interpolate, using the resulting three differences only —

-1

0-985612

+0-014388

-0-028776

-0-022749

-0-014388

-0 051525

—

0-985612

-0 065913

—

0-919699

—

Interpolating for the value of we have —

+ ^^^j=1-5Ao^+0*375Ao2-0*0625Ao^

= 0-997825

The true value, as stated above, is 0*997839, and we have got a closer
result by this rearrangement, using third differences only, than we did by
using nine differences before.

24.12 Possible forms of polynomials , — The student may also get into
difficulties if he does not bear in mind the forms that polynomials can,
and cannot, take ; and if he attempts to use this method of interpolation
where the polynomial is unlikely to represent the function well even over
a moderate range. A polynomial (parabola) of the second order can take
only the form [a) in fig. 24.1 . A polynomial of the third order can take the
form (6), or the form (c) with a wave in the centre. A polynomial of the
fourth order can take a form very much resembling (6), but flatter in the
centre, or a form like {c)y but with three instead of two half-waves in the
middle ; and so on. A polynomial cannot take the form (1) of a curve
tangential or asymptotic to the vertical, like the end near zero of an ideal
frequency-curve of the distribution-of-wealth type, or (2) of a curve
slowly dropping asymptotically to the horizontal, like a logarithmic curve
or the tail of the normal curve — and such functions, mathematical or
empirical, are very frequent in statistics. In this latter case it would be
more probable that the function could be represented by a function of the
form

y = . . .

Then taking logs we have —

U =l0gey * • *

that is to say, we come back to the polynomial. Hence, if the function
we are dealing with is tailing slowly away to zero, it is probably best to
take logarithms and then interpolate on the logarithms. That is why in
Example 24.4 we carried out a check in that way. There, as it happened,
the direct method did not lead to bad results, but it is quite possible for it to
give a completely nonsensical answer. For example, at the extreme end

INTERPOLATION AND GRADUATION 567

of the table for j^=28 (^'=29), we are given only the values of P
corresponding to the following values of —

0*066128

-.0*059661

4-0*053601

-0-047929

0*006467

-0*006060

+0 005672

—

0*000407

-0*000388

—

0-000019

—

Taking differences as shown and interpolating to get an estimate of the
value of P for x^=^^> he. %. 5 , we have —

= ^^0+1 •5Ao^+0-375Ao2-~0*0625Ao3
= -0*000268

But this is nonsense, for P cannot be negative. The polynomial has done
its best ; it reproduces the values at 40, 50, 60 and 70 — ^but it can only do
this by taking a form like {c) of
fig. 24.1 (reversed) with a wave in
the centre. It -has, as a matter of
fact, a minimum at x^ =56 • 6 and a
maximum at ;^2=65*8, or at 1*66
and 2 • 58 on the scale of u's with 40
as zero and 10 as the unit interval.

If, instead, we take logarithms
of the above values of P, inter-
polate to third differences and then
convert back to numbers, as in
Example 24.4, we find 0*001699
for the required value of P — a
value which is rational and is
probably not far from the truth.

For x^=30, P =0*363218. Even
bringing in this much larger value
and using logarithmic interpolation with four differences, we find 0*001746
for the value of P at This suggests that at least we may trust

the value to two figures as 0*0017, which would be sufficient for practice ;
but the value has not been checked by direct calculation.

Effect of errors in ^ on the differences

24.13 The student may notice and be troubled by the fact that, in
the Normal Curve Tables in the Appendix, second ciifferences appear to
get a little irregular towards the tail of the curve ; the phenomenon will
become much more evident if he continues the second differences rather
further than they have been entered, and still more so in the higher differ-
ences if he proceeds to write them out. The irregularities in question are

568

THEORY OF STATISTICS

due solely to the errors of rounding of in the last decimal place of the
function. Before proceeding to consider the total ef ect of such a system
of errors it may be best to consider the efect of a single error.

24.14 Effect of an error in a single value of u . — If u=v~\-w,
and so on for all orders of diferences. Hence, if v represents the true
value of u and w represents an error, the diferences of the error will
simply be superposed on the diferences of u, and we may consider the
former by themselves. We may then, as below, take the true values of u
as zero, and insert an error only at one point, say -\-e.

A®

A«

+ e

— 6 (?

+ e

- 5^

+ 155

-f e

—4e

-205

+ e

-lOe

+ 155

-2e

—4e

+ 5e

— 65

—e

+ e

— e

+ e

— e

+ ^

The resulting differences are written down above, up to those of the sixth
order, and it is evident that the numerical coefficients of e in the diferences
or order r are given by the terms of (1— 1)^ The efect of the initial
error is therefore very rapidly increased as we proceed to higher and higher
orders of difference, especially after the first three differences are past. An
error of in u can produce an error of +3^ or —Ze in the third differences,
of in the fourth differences, of lOe in the fifth and of 20e in the sixth.
The maximum numerical coefficient for order r is derived from that for
order 1 by multiplying the latter by 2 if r is even, or by 2r /{r+l) if
r IS odd.

This magnification of the error renders differencing a very useful
method of checking the calculated table of a function, and it is often
employed for that purpose. The matter is not quite simple, for the effects
of errors of rounding off in the last decimal place will be superposed on the
effects of any actual mistake, but nevertheless the effects of the mistake
are likely to show themselves clearly in, say, third or fourth differences.
In the following table of square roots, for example, nothing is obviously
wrong, but an error of 2 units in the last place has been introduced into the
square root of 1 5, which should read 3 * 87298 (or more precisely, 3 • 8729833) .
When we proceed to take differences, however, a suspicious irregularity
shows itself in the third differences, and in the fourth differences it is clear
that something is wrong. Since the position of the peak '' rises half a
line at each differencing, the peak +2 shows that the mistake is in the
root of 15. We can even estimate the magnitude of the error. If the fifth
differences may be taken as approximately constant, we ought to get a fair
estimate of the true fourth difference at the peak +2 by adding together
that difference and the two on each side of it, the total effect of the error

INTERPOLATION AND GRADUATION

569

Number

Square root

A‘(+)

AM-)

AM + )

3-16228

0-15434

686

-14

3-31662

0-14748

603

-12

3-46410

0-14145

534

-14

3 60555

0-13611

477

-f 2

3-74166

0-13134

434

-14

3 87300

• 0-12700

389

0-12311

358

- 6

1 ;

4-12311

0-11953

327

—

4-24264

0-11626

302 '

—

4-35890

0-11324

—

4-47214

—

— —

e thus averaging out — compare the scheme showing the effect of the single
error given above. This average is —7*6. We then have —

Qe = + 2 -(- 7 - 6 )

(2 = + 1-6

This is very near the correct value^ which, as will be seen from the true
value of the root stated, is 300—298-33 or 1-67, the unit in the column
being the last place of decimals of the function.

24.15 Effect of a series of random errors in u . — Suppose these errors
to be a, by Cy dy Cy as below. Writing down their differences, we have the
following results —

Error

b—a

c—2b-\-a

d — 3c "j” 36 — d

4c? -f 6c— 46;}- a

c—h

d — 2c-|-&

e — 3(i-j-3c— 6

—

d—c

e—2d\-c

—

e—d

—

The general result is obvious.

In differences of the rth order, the resultant

error in any one difference is the sum of r + 1 of the original errors multiphed
in succession by the terms in the binomial expansion of (1—1)^ or is
of the form

-re^-

r{r-l)(r—2)^

1.2

1.2.3

( 24 . 6 )

If the errors e are distributed in a purely random way, so that Cjc is un-
correlated with Cj^j^^sy and if it may be assumed that the mean error is zero,
then the mean error in the difference of the rth order will also in a long
series tend to zero, and the standard deviation, of the above quantity
(24,6) is given by

= F{r)s,^

. (24.7)

570

THEORY OF STATISTICS

where Sq is the s.d. of the original errors e, and F{r) is the sum of the squares
of the terms in the binomial expansion of (1 —1)'. This may be shown to

be equal to

F{r) increases very rapidly with r. The following table gives the value
of F(r) and of its square root from ^=1 to —

F{r)

1-41

2-45

4-47

8-37

252

15-87

924

30-40

The standard deviation of errors in the fourth dif erences is therefore over
eight times, and in the sixth differences over thirty times, the s.d. of the
errors affecting u.

If the decimal place in u be regarded as following the last figure
retained, the errors of rounding off that figure may be regarded as uniformly
di stribu ted over a range ±0-5, and their standard deviation, Sq, is therefore
Vl /12 or 0-288675. This gives the following figures for the s.d. of errors
in the successive orders of difference owing to the errors of rounding off

m u —

Order of difference S.d of errors

0-41

0-71

1*29

2-42

4-58

8-77

The effect of the errors of rounding off evidently increases very rapidly
with the order of difference. With a mathematical function for which
the true differences rapidly and continuously converge, the effect of the
errors will in fact soon, so to speak, take charge ” ; the observed differ-
ences will rapidly and steadily diverge, growing larger with each successive
differencing. At the same time two other phenomena will show them-
selves. Looking back at the scheme showing the effect of the errors
h, c, d, e, it wiU be seen that in any one column the same error enters
into successive differences with sign reversed. Also in any one line
the same error enters into successive differences with sign reversed.
Hence, as the effect of errors of rounding off becomes overwhelmingly
great, (1) the differences of the same order tend to alternate in sign, (2)
differences of successive orders on the same line tend to alternate in sign.
If these phenomena start to show themselves, the student may well
suspect he has gone too far in his differencing. It is evidently no use
proceeding to an order of differences mainly significant of errors.

These results for the effect on differences of a random series of errors
have an application, not only to the effect of errors of rounding off in
mathematical tables, but also to the theory of the variate-difference
method (26,31).

INTERPOLATION AND GRADUATION

571

Effect on differences of subdividing an interval

24.16 We mentioned early in this chapter (24.2) that, in general, it
would become possible to use simple interpolation alone on a table of
a mathematical function provided intervals were made sufficiently fine,
but this was not proved. Let us consider the effect on the differences
of subdividing an interval ; it will suffice to take the case of halving it,
and for brevity let us confine ourselves to the first three differences.

In terms of Newton's formula the values of at 0, 0*5, 1, 1*5, are

“^0 — ^0

0 5 0 ^ 0 0 -r I . . (24.8)

% = %+Ao^

SAqI +0 • 375Ao2 -0 • 0625Ao®

If the student will write down these expressions at the left of a sheet
of foolscap placed lengthwise, and take the differences in the ordinary
way, he will find that the new leading differences for the subdivided
series with intervals of half the original interval are given by

V = 0-5Ao^-0*125AoHO*0625Ao®
(^q2^0-25Ao^--0-125Ao®

V =0*125Ao^

(24.9)

If the A's of the original series converge rapidly, an assumption really
implied by the fact that we stopped at the third difference, so that we
can regard the successive A's as of different orders of magnitude, it will
be seen that is of the order of magnitude 0 • SAq^, ^ 0 ^ is of the order ot
magnitude 0*25Ao^, and of the order of magnitude 0*125A(j^. That
is to say, the new differences are not only smaller than the original
differences, but converge much more rapidly.

If we had divided the original interval into ten instead of only two
parts, we could have found the new leading differences in precisely the
same way, and would then have obtained the result that was of the
order of magnitude OTAqS of the order of magnitude O-OlAp^ and
so on, the general rule being obvious. Hence it is only necessary to
subdivide the interval sufficiently in order to render the differences so
rapidly convergent that first differences alone can be used.

In works on the method of differences, tables will usually be found
giving for various values of the number of subdivisions the formulae
relating the d's to the A’s.

We now turn to some statistical problems.

Breaking up a group

24.17 Suppose we are given the numbers living, or the numbers of
deaths, in successive ten-year age-groups, we may often desire to estimate

57 ^

THEORY OF STATISTICS

the numbers in smaller, e.g. five-year, age-groups, or even at single years
of age. The initial difficulty and the method of procedure will best be
shown by an illustration.

Example 24.5

The following are the numbers of

deaths in four successive ten-year

age-groups. Required to estimate the numbers of deaths at 45-50 and
50-55.

Age-group

Deaths

25^

13,229

35-

18,139

45-

24,225

55-

31,496

Now evidently interpolating directly between these figures will not help
us. If we interpolated directly between the figure for 35- and the figure
for 45- (half-way between), we would only have an estimate of the numbers
in the ten-ytdx age-group 40-50. We must proceed as follows. Add
up the given numbers step by step ; this will give us a new set of figures
showing the numbers over 25 but less than 35, over 25 but less than 45,
over 25 but less than 55, and over 25 but less than 65. Interpolate in
this new series to find the number over 25 but less than 50, and the differ-
ences from the numbers next above and below will give the answer
desired. The work is as follows —

Exact age

Sum of deaths
from 25 to age
stated

A®

+ 13,229

+4,910

+ 1,176

13,229

+ 18,139

+6,086

+ 1,185

31,368

+24,225

+ 7,271

—

55,593

+31,496

—

87,089

—

Column 2 gives the numbers from age 25 up to each age stated ; column
3 the first differences, reproducing the numbers in the age-groups ;
columns 4 and 5 the second and third differences. Since the two third
differences are very nearly equal, working to third differences ought to
give us a very fair result. We can accordingly take age 35 as our zero,
and age 50 will be 1 *5 on the scale with the interval as unit. We have
accordingly,

= Wo+l‘5Ao^+0-375Ao2-0-0625Ao2
= 13,229+1 -5(18,139) +0-375(6,086) -0-0625(1, 185)

= 42,645-7

INTERPOLATION AND GRADUATION

573

or 42,646 to the nearest unit. Subtracting 31,368 from 42,646, and
42,646 from 55,593, we then have for our estimates of the numbers of
deaths —

45-50 11,278
50-55 12,947

As a matter of fact, the numbers in quinquennial groups were given, and
for 45-50, 50-55, were actually 11,404 and 12,821 ; the error of our
estimates accordingly is only of the order of 1 per cent.

Example 24.6. — From the same data, estimate the number of deaths
in the year of age 50-51.

The limits of this group on our scale of intervals are, with 35 as origin,
1*5 and 1 *6. We have already found the number up to 1 *5 in Example
24.5, and it remains only to determine the number up to T6, the difference
between the two figures then giving the answer sought —

= Wo-|-l-6AoHO-48Ao2-0*064Ao^

= 1 3,229 + 1 • 6(18, 1 39) +0 • 48(6,086) -0 • 064 ( 1 , i 85)

= 45,096*8

or 45,097 to the nearest unit. Hence the answer is 45,097 —42,646, or
2451.

Simple formula for halving a group

24.18 The problem of estimating the numbers in the two five-year
groups of which a ten-year group is composed occurs so often, that it is
worth while deriving a simple second-difference formula for the purpose.
Let u's denote numbers in five-year groups, w's numbers in ten-year
groups ; and let <^'s and A's denote the corresponding differences. For
second differences we need only consider three consecutive ten-year groups.
From Newton’s formula we have —

Wq = Uq

= 2%-hV

% = % -1-450^ -j- 6^0^
= 1^0 * 4 - 5 ^ 0 ^-}- 10 ^ 0 ^

574

THEORY OF STATISTICS

Now write down these values of the w’s and difference —

X Wx

0 2m„+V 4V+4V 8V

1 2«„+5V+4V 4V+12V

2 2«o+9V+I6V

Whence

Ao^ = 4(doM-^o")

Ao^ = 8 ^ 0 ^
or

- iAo^

V -

Hence,

= %+4Ao^—

= -~*( 2 Aoi+Ao«)

It will be convenient for practical work to express this directly in terms
of the w*s —

2Ao^ = 2wj^^2w^

Ao^ = W2—2wi+Wo

2Ao^+Ao^ =

Whence finally,

^2 == M^i+K^o—^ 2 )} • • • (24.10)

Thus, taking the figures and problem of Example 24.5 again, we have —

Wo = 18,139
Wi = 24,225
= 31,496

i{wQ—w^) == 1,669-6

= 24,225

and half this gives

22,555-4

= 11,278

to the nearest unit, as before. For Wg, of course, we have also, as before,
24,225—11,278=12,947. Equation (24.10) is really equivalent to the

INTERPOLATION AND GRADUATION

575

method of Example 24.5, though in that illustration we used three differ-
ences. But the third differences of the numbers ‘‘ aged over 25 but
under ” are equivalent to the second differences of the numbers m the
successive age-groups.

Graduation

24.19 If a graph is drawn showing the numbers of either sex living
at each single year of age, as given in any census which provides data in
such detail, it will be found anything but smooth, showing the oddest
peaks and hollows which repeat themselves, once adult life is reached, at
ages showing the same final digits. Thus, in the Census of England and
Wales there are conspicuous peaks at the round-numbered ages 30, 40, 50,
etc. (last birthday), and hollows or deficiencies at the ages ending with 1
and, less emphatically, at the ages ending with 7. With returns from less
educated populations, the phenomenon may become almost ludicrous, e.g.
m a certain Indian census sample-count —

Age last birthday

Number of males

927

12,294

652

2,058

672

892

7,723

1,437

870

1,362

467

10,391

460

Now whatever irregularities might occur in the true figures, we may be
quite certain that they should not show errors that are simply a function
of the final digit of the age. We would prefer, therefore, to eliminate these
errors. We could do so, somewhat roughly, by drawing a graph as
suggested and sweeping a clean curve through the rather scattered and
irregular points given by the data, subsequently reading off smoothed or
graduated figures from the curve. The graphic process has many points to
recommend it, but is very dependent on personal skill and judgment. It
would be convenient to use a more mechanical " process that anyone
could apply and be sure of obtaining the same results if he used the same
process. It would be quite possible to fit polynomials to the data by the
methods of Chapter 15, but this would in general entail a great deal of
labour and would not necessarily lead to satisfactory results, e.g. with such
highly erratic data as those above. More suitable processes can be

576

THEORY OF STATISTICS

founded on the method of differences, and the general idea of them all it
quite simple, though the details may vary greatly and the practical working
of some of them become rather complex. All methods begin by assuming
that the totals of certain age-groups — five-year or ten-year age-groups as
a rule — are reasonably accurate. These totals can then be redistributed
over single years of age by the elementary process of Examples 24.5 and
24.6, or the procedure can be in some way elaborated. We shall illustrate
only the simple process.

Example 24.7. — The English Census of 1911 gives the following numbers
of males in the three age-groups stated. Obtain graduated numbers at
single years of age for the decade 40 to 49.

Age-group Number

30- 2,637,304

40- 2 001,178

50- 1,376,236

As before, we form the sum of these numbers step by step from the
top and then take differences.

Exact

age

Sum of
numbers
from 30

A'(+)

A“(-)

A’{ + )

2,637,304 i

636,126

11,184

2,637,304

2,001,178

624,942

—

4,638,482

1,376,236

—

6,014,718

—

We now, taking 30 as our zero, require to interpolate at 1 • 1 , 1 • 2, 1 • 3, etc,
to 1 • 9. The coefficients of the several differences in the successive applica-
tions of Newton's formula are — .

•+■1*1

+0*055

-0 0165

+ 1-2

+0*12

-0*032

-1-1-3

+0*195

-0*0455

-fl-4

+0*28

-0*056

+ 1-5

+0*375

-0 0625

-fl-6

+0*48

-0*064

+ 1-7

+0*595

-0*0595

-fl-8

+0*72

-0*048

-fl-9

+0*855

-0*0285

The results, with the known numbers to age 40 and to age 50 added,
are as given in the second column below, and in the fourth column they
are differenced to obtain the graduated numbers at each year of age, the
total of which must agree with the observed total in the ten-year group.

INTERPOLATION AND GRADUATION

577

Exact

age

Sum of population
from 30 to age
stated

Age

last

birthday

Graduated

number

2,637,304

228,559

2,865,863

222,209

3.088,072

215,870

43 1

3,303,942

209,542

3,513,484

203,226

3,716,710

196,920

3,913,630

190,626

4,104,256

184,344

4,288,600

178,071

4,466,671

4,638,482

171,811

Total

—

2,001,178

Below, these figures afe compared with the actual returns at the single
years of age and with two other graduations : (1) A graduation given in
the Census report and prepared by Mr. George King, F.I.A., based on
certain quinquennial age-groups. (2) A graduation using analogous
methods, but based on ten-year age-groups, made at a later date in the
Government Actuary's Department, and reproduced by permission. The
methods are described in rather more detail below.

Age

last

birthday

Census

numbers

Graduation

above

King’s

graduation.

Graduation

262,690

228,559

231,070

231,397

198,344

222,209

223,721

225,456

226,889

215,870

216,556

219,233

196,204 1

209,542

209,314

212,785

190,949

203.226

202,143

206,169

202,458

196,920 ;

195,193

199,442

184,881

190,626

188,610

192,661

176,713

184,344

182,577

185,883

189,271

178.071

176,994

179,165

172,779

171,811

171,589 i

172,564

Total

2,001,178

1,997,767

2,024,755

If we compare the closeness of fit of the several graduations to the
Census returns by adding up the differences, observed number less gradu-
ated number, without regard to their sign, and expressing this total as a

578

THEORY OF STATISTICS

percentage of the population (2,001,178), it will be found that our gradua-
tion gives a percentage deviation of 6-28, King*s graduation {K^ a per-
centage deviation of 6*09, and the graduation a percentage deviation of
6-40 — figures which do not differ very largely. It will be noticed, how-
ever, that both the K graduations give, over the range considered, a small
biased error, the total population over the ten years being too small for
Ki and too large for regards the deviations of the several gradua-

tions from one another, the percentage deviation of our graduation from
is 0 • 64 and from reckoned in each case on the true total popula-

tion, and the percentage deviation of from is 1 • 35, reckoned on the
total. At some individual ages the differences run up to nearly 2 per
cent. This is a warning to the student that while it is true that the use
of any one of these methods by different workers must, unlike the use of the
graphic method, lead to the same result, yet the choice of different methods
may lead to results almost, if not quite, as divergent as those obtained by
different users of the graphic process. Graduated numbers of hundreds of
thousands carried to the last unit suggest a degree of precision much
higher than exists.

There is evidently a certain imperfection in the elementary method we
have used. If we employed the same method to graduate the numbers at
ages 30 to 39, using the numbers in the three ten-year age-groups 20-, 30~,
40-, there would be a discontinuity at 40, for the two graduated series would
be given by arcs of distinct polynomials. The discontinuity might not
be conspicuous, but it would be there and would probably be brought out
by differencing. To get over this, at least in part, a simple adjustment
can be used. Continue the graduated series for 30 to 39 over the next few
years of age, say to 42. Also continue our series for 40 to 49 backwards to
37, Over the six years 37 to 42 we then have two graduated values at
each age, and these may then be averaged with weights which gradually
throw the weight from the earlier series on to the later — say such simple
weights as 6 to 1 , 5 to 2, 4 to 3, 3 to 4, 2 to 5, 1 to 6. We have also paid no
particular attention to the choice of the limits of our ten-year age-group.
Of course it might happen that the numbers were only compiled in ten-
year groups like 20-, 30-, 40-, etc., and then there would be no choice.
But if the figures are given at single years, the choice is at our disposal,
and it may be that we have not chosen wisely. Part of the excess at the
peak figure is probably drawn from lower ages, and it might have been
better to keep the “ peak at the round-number ages well inside the group,
e.g. by compiling totals for the decades 35-, 45-, etc., rather than those
used.

King, in the Census graduation, used five-year age-groups as his
basis, and chose the limits 4-8, 9-13, 14-18, etc., as probably giving the
totals nearest the truth. Taking these five-year totals in successive sets
of three, he used the precise procedure of our Example 24.6 to determine
a graduated figure for the central year of the fifteen — e.g. the three groups

Interpolation and graduation

579

covering ages 4-18 would give a graduated number at age 11, the three
covering ages 9 to 23 would give a graduated number at age 16, and so
on. But here his process broke away. Taking four consecutive graduated
numbers five years apart and determined m this w^ay as " pivotal values,"
he used the method of differences to determine a polynomial of the third
order not passing through the four points u^, but subjected to

the four conditions (1) that it should pass through the two points
and (2) that at and U 2 it should have a common tangent with the
corresponding arc determined from the next (overlapping) set of pivotal
values. In this way continuity was assured, but equality of observed
and graduated totals for the five-year groups was lost. (The process
used was a simplification of the process of osculatory interpolation, by which
two arcs meeting at a point are given not only a common tangent but also
a common radius of curvature. It might be called “ tangential inter-
polation.") The desirability of using five-year groups may be questioned.
It is true that ten-year groups are rather large, but the errors that we are
trying to eliminate are definitely functions of the ten final digits, and
however the limits are chosen there is likely to remain a systematic
difference between the adjacent groups of successive pairs if five-year
groups are used.

The test of in which an analogous process was used but based
on the ten-year age-groups 5-14, 15-24, etc., was therefore of interest.
Over the range of 30-80 years the differences between and gave a
smoothly running cyclical curve with a tendency towards a period of
ten years, as might have been expected.

The simple process given in Example 24.7 is applicable throughout
the bulk of life, but not at the two ends of the series, where special tricks
of the trade have to be employed. The difficulty of interpolating in a
“ tail," where the numbers are slowly approaching zero, has already been
pointed out. For graduation these difficulties are increased, and it is
often best to drop the method of differences altogether and use some
special process, such as assuming a law of decrease or fitting the tail of a
frequency-distribution.

Inverse interpolation

24,20 By interpolation we determine the value of the function for a
given value of the variable. If we are given the value of the function
and find the corresponding value of the variable, we are performing
inverse interpolation. The student has carried out the process, in a form
corresponding to simple interpolation, whenever he has determined the
number corresponding to a given logarithm by the use of a table of
logarithms — not a table of antilogarithms. If we need only take first
differences into consideration, the process is, in fact, very simple. From
Newton's formula we have

58 o

whence

THEORY OE statistics

X =

. (24.11)

where % will naturally be taken as the tabulated value next below %.
If we must take second differences also into account, we have

which gives the quadratic for x

or, solving,

2Ao^ ^0^ I '^o) I /2Ao^

2Ao2 Ao^ 2Ao2* ;

(24.12)

(24.13)

The sign to be taken for the square root will be evident on carrying out
the arithmetic.

This is not always a very convenient expression to use, the solution
(compare Example 24.8 below) being given as a comparatively small
difference between two large quantities. If % is the approximate solution
given by first differences, we can replace a; in equation (24.12) by
and solve for the correction h on the assumption that may be neglected.

This gives

l. _

2;eiA2+2Aoi-Ao*

2+(2a;i — \)p

, (24.14)

where

A 2
p

' A ^

. (24.15)

If we may further assume that p is small, this reduces to

h = \XjX\-x^)p .

. (24.16)

Obtaining a first approximation from first differences, we can use (24.16)
to get a second approximation, then insert this second approximation in
(24.16) and get a third approximation, and so on until the process of
approximation makes no 'further difference. But note the assumption
made that p is small.

Example 24.8

To find the approximate value of the quartile deviation, i.e. the value
of xja for which .4=0 *75, m the normal curve, given that for
= 0*6, 0*7, 0*8 the values of A are respectivelv 0*72573, 0*75804,
0-78814.

INTERPOLATION AND GRADUATION

581

The data are —

X la A Aq^

0^6 0*72575 +0*03229 -0*00219

Hence,

Uo^—Uq = 0*02425

and the first approximation to x by first differences only is

, 0*02425 , n • X

= +0-07510

or measured from the zero of the scale, the first approximation to the
quartile deviation is 0-67510.

Turning now to the quadratic (24.13), the solution is

X = 15-2443-14-4997

= 0-7446 interval

= 0-07446

the sign of the root having evidently to be taken as negative. Using
second differences, then, oj^ approximation to the quartile deviation is

0-67446

The true value to five places is

0-67449

so the use of second differences only has left an error in the last digit.

Let us see how the suggested process of approximation would have
worked. From (24.16) —

h = -0-0339114 x 0-751x0-249
= -0-00634
Xj = 0-751

Xj = 0-74466

Now taking Xg as the second approximation —

h = -0-0339114 x 0-74466 x 0-25534
= -0-00645
Xi = 0-751

X 3 = 0-74455

If we repeat the same process again, x^ =0-74455, which is the same as x^,
so it is no use going further, and 0-67446 is as close as we can get.

582

THEORY OF STATISTICS

If third and higher orders of difference are brought into account, we
have an equation of higher degree than the second, which can be solved
by Newton’s method of approximation, but the student will find more
direct methods given in advanced works.

Estimation of the position of a maximum

24.21 In this and the following problem an elementary knowledge of
the calculus is assumed ; the student who does not know the calculus
may nevertheless find the results useful.

Suppose we are given three equidistant ordinates Uq, u^, Uo at 0, 1
and 2. Required to find the position of the maximum of the parabola
passing through the tops of the ordinates. We have —

Differentiating with respect to x and equating to zero, the abscissa of the
maximum is given by

Ao^+i(2^-^l)Ao2 = 0
or

^=0-5-V .... (24.17)

Very often, perhaps most frequently, our data are not ordinates but
rather areas ; e.g. if we want to estimate roughly the position of the mode,
our data will be the total frequencies in three successive class-intervals —
not the central ordinates of those intervals. We should then, as in Example
24.5, form the sum of these data step by step and take the second differential
of the polynomial passing through the resultant points in order to deter-
mine the mode. Thus, calling the sum w —

Sum w

- 0-5

Wo+Ao^

4 - 0*5

«o-i- 2 Ao^+Ao*

+ 15
+ 2*5

2 wo+Ao^

3 wo+ 3 Ao^+Ao®

It must be remembered that the sum w starts at half an interval bel
zero, as shown. Using S's to denote the differences of w —

== ®o+^Mo-

dhi'^

dx^

V = Ao^

x{x-\){x-2) ,

— 6 ^0

^Ao‘+(^-l)Ao^ =0

INTERPOLATION AND GRADUATION

583

Since x is now measured from — this is the same answer as before. If
we are concerned only with second differences of the data, and not with
differences of any higher order, it does not matter whether our data are
ordinates or areas.

The method must be used with caution ; obviously it cannot give at all
a precise result unless the data run smoothly, and if it be used for determin-
ing the mode, may easily give an answer appreciably divergent from that
obtained by fitting a frequency-curve. The following illustration will serve
as a warning —

Example 24.9. — The following are the frequencies near the mode in a
distribution of barometer heights. Estimate the position of the mode, (1)
from the first three, (2) from the last three.

Differencing —

Height (inches)

29- 9

30- 0

30-1

30-2

Frequency

339-5

382*5

395-5

315

Height

(inches)

Frequency

29-9

339-5

+43

-30

30-0

382-5

+ 13

—93-5

30*1

395-5

-80 -5

30-2

315

—

Taking the first three frequencies and their differences —

= 0*5+g^ == 1-933 intervals == 0-193 inch
Estimated mode = 30*093

Taking the second three frequencies and their differences —

X = = 0-639 interval — 0*064 inch

Estimated mode = 30*064

Our two answers therefore differ sensibly from each other, and also
from the value given by a fitted Pearson curve, viz. 30-039.

Modifying central ordinates to equivalent areas

24.22 Supposing we fit a theoretical frequency-curve to an actual
distribution, and want to determine the goodness of fit " by the
method. We would usually proceed by calculating, from the curve

584

THEORY OF STATISTICS

determined, the ordinates at the centre of each class-interval and taking
these as the frequencies. But this procedure is not exact, for the central
ordinates are not precise measures of the areas. In a class-interval
centred exactly on the mode, for example, the central (maximum) ordinate
obviously gives too large a value for the area. Required, to obtain some
simple formula for modifying the central ordinates so as to give the areas.

We have, by Newton’s formula.

Integrate this expression for the interval round i.e. between the
limits 0*5 and 1-5, and we will have an expression for the equivalent area,
say —

Jo 5

= %+Ao*^+^Ao^

. (24.18)

The first form of the formula is, in general, the more convenient, but the
second may be the better if correction is wanted only to a single value of u.
Example 24.10

Table 24.5 (page 585) gives in column 2 the calculated ordinates of a
Pearson curve at the centres of the class-intervals. In columns 3 and 4
are given the first and second differences, and in column 5 are given
the corrections /24, shifted one line down so as to be on the same line
as the ordinate to be corrected. Finally, in column 6 we have the sum
of the ordinate and the correction, or the area. The totals given at
the foot are simply for the purpose of checking , since columns 2 and 3
both begin and end with zero, the sums of both first and second differences
must be zero. Since column 5 is derived from column 4 by dividing
by 24, its sum should also be zero, but errors of rounding off have made
a very small negative excess. All the corrections are very small ; they
are necessarily greatest where the curvature is greatest.

24.23 A few words in conclusion. The process of interpolation, and
still more that of graduation, is almost as much artistic as scientific. No
absolute rules can be laid down, judgment must be used, and it is the
experienced craftsman who is likely to get the best results with the least
labour. If the student turns up his Latin dictionary he will find that
inter polar e means not only '' to polish up ” {polire, to polish) — so that
graduation is really the implication of the word — but hence to corrupt,
to falsify.” It will do him no harm to bear this etymological meaning in
mind, and keep a look-out accordingly.

INTERPOLATION AND GRADUATION

585

TABLE 24.5

Class-

mterval

Central

ordinate

Correction

Area

0-00

0-08

0-00

+ 0-08

0-70

+0-00

0-00

0-08

+ 0-78

3-08

+0-03

0-11

0*86

+ 3-86

6-91

+0-13

0-99

4-72

+ 10-77

7-18

+0-29

5-01

15-49

+ 17-95

0-55

+0-30

15-79

33-44

+ 17-40

10-76

-0-02

33-42

50-84

+ 6-64

—

13-70

-0-45

50-39

57-48

- 7-06

7-88

-0*57

56-91

50-42

-14-94

0-06

-0-33

50-09

35-48

-14-88

4-37

+0-00

35-48

20-60

-10-51

4-67

+0-18

1 20-78

10-09

— 5-84

3*15

+0-19

10-28

4-25

— 2-69

1-64

+0-13

4-38

1-56

— 1-05

0-69

+0-07

1-63

0-51

- 0-36

0-25

+0-03

0-54

0-15

- 0-11

0-08

+0*01

0*16

0-04

- 0-03

: +

0-02

+0 • 00

0-04

0-01

. - 0-01

0-01

+0-00

0-01

0-00

+0*00

0-00

Totals

1 286-02

+57-48

-57-48

+32-89

-32-89

+ 1*36
-1*37

286-01

SUMMARY

1. The first, second, third, . . . differences of a function are defined
by the equations

V =

etc.

the intervals between successive values of the variable x being equal

2. By means of Newton’s formula.

we can interpolate for the value of

586

THEORY OF STATISTICS

3. Errors in the values of u become of increasing importance as the
order of the differences increases.

4. For inverse interpolation

_ U^-Uq

^ ” A 1
ao

for first differences ;

_ 2Aoi-Ao2 72 Ao1-Ao2\2

2Ao2 \ 2 Ao« /

for second differences.

We can also proceed by successive approximation. If is the approxi-
mate solution by first differences, a closer approximation is %+A, where

Xi{l

2+(2^i-l)^

EXERCISES

24.1 Given the following values for the normal integral

xjtj

1-4

•91924

1-5

•93319

1-6

•94520

1-7

•95543

find the value of A for ^ /a ==1-54, noting the successive approximations
up to third differences. Take % at 1 ’4.

24.2 Find as closely as possible the value of P for x2==ll'7 from the
following entries in the table [Tables for Statisticians) : >^=17 (^'=18).
Note the successive approximations and the number of places to which
your final answer is probably trustworthy.

0*903610

0*856564

0*800136

0*736186

INTERPOLATION AND GRADUATION 587

24.3 From the following entries in the same table for r— 24(?t'=25),
estimate as closely as you can the value of P for Similarly,

estimate the closeness of your approximation.

0-184752

0-021387

0-001416

0-000064

24.4 The following table gives the deaths of males registered in
England and Wales during the three years 1930, 1931, 1932, at the ages
stated. The figures on the right give the totals of the quinquennial groups
which were, on this occasion, held to give the best totals for determining
quinquennial “ pivotal values.*' Find graduated numbers for the ages
40 to 44 inclusive.

Age

Numbers

Qumqueanial totals

3394

3505

3501

3947

3998

18,345

4220

4281

5024

43 !

4993

44 !

5260

23,778

5998

6113

6463

6921

7663

33,158

24.5 Let Ui, «< 2 , . . . be the numbers in fifteen consecutive years of
age, as in Exercise 24.- , and Wq, the totals in the three quinquennial

groups. Show that if we want only the graduated figure for as a
pivotal value/* this may be written down at once from the equation

= 0 ' 2 %— 0 * 008 A%o

(King*s formula). Verify by comparison with your answer to Exercise 24.4,

588

THEORY OF STATISTICS

24.6 Generalising the above result, show that if Wq, are three

successive age-groups of r years each, we have for the graduated central
value 9 ^ / \

^8f-l ^ «'r

2 r 24r^ \ ^ /

and hence if r become indefinitely great, the central ordinate of the middle
group of three, with areas Wq, w^, and common base c, is given by

c 24 \ c /

Verify by finding approximately the central ordinate of the normal curve
from the areas between —0-3 and — 0-1, —0-1 and +0*1, +0-1 and
+0*3 xfo.

24.7 From the following (abbreviated) entries in the table, v=9
(n'=10), estimate the value of x^ for which P=0*25 —

0 2757

0-2133

0 1626

24.8 The next table shows a frequency-distribution of 1,000 observations,
and also gives the frequencies summed from the top. Estimate (1) the
median, (2) the first decile, (3) the ninth decile, {a) as usual by simple
interpolation, (b) by bringing second differences also into account.

Interval

Frequency

Sum of
frequencies
from 0 to

O-I

1-2

104

2-3

114

218

3-4

141

359

4-5

158

517

5-6

142

659

6-7

119

778

7-8

873

8-9

936

9-10

969

10-11

987

11-12

995

12-13

: 13

997

13-14

999

14-15

—

999

15-16

1000

Total

1000

—

iNTfiEtOLATlON ANt) GRADUATION

589

24.9 The following are the mean temperatures (Fahrenheit) at Greenwich
on three days 30 days apart round the penods of summer maximum and
winter minimum. Estimate the approximate dates and values of the
maximum and minimum.

Day

Date

Temp.

Date

Temp

15th June

58-8

16th Dec.

40-7

15th July

63-4

15th Jan.

38-1

14th Aug.

62-5

14th Feb.

39*3

24.10 Taking the value of the central ordinate of the normal curve from
Appendix Table 1, estimate the area between the limits ±0*l;t;/(T, and
verify your answer from the area table.

CHAl lER TWENTY - FIVE

INDEX NUMBERS

The general problem

25.1 It often happens, particularly in economic statistics, that a set of
similar events moving through time or space gives rise to some general
concept expressing variation in their common element. The prices of a
number of commodities on sale lead to the notion of a relative price
level ; the various outputs of manufacturing plants generate the idea
of changes in the “ volume of industrial production as a thing-in-itself ;
the yields of different crops in a set of agricultural districts suggest a
comparison of “ agricultural productivity between different geographical
areas. Although there is room for argument about the role of some of
these concepts in providing explanations of phenomena, it will not in
general be denied that they are useful subjects of inquiry, and in particular
that knowledge is advanced when we can measure the properties which
they represent, or at least the relative values at different times and in
different places. In fact, when we leave the domain of philosophical
discussion some of these concepts assume a degree of practical importance
which is denied to more concrete and less contentious ideas ; whether we
agree or not that there is such a thing as the cost-ob living, we must
admit that movements in wages and salaries in many countries are
influenced (and in some are determined) by a measure of the relative
level of the ''cost-obliving*' expressed in the most definite numerical
terms.

25.2 In this chapter we shall be concerned with the measurement of such
concepts as relative price-levels and changes in the general price-level
by means of index-numbers, i.e. numbers which tell us, or at least purport
to tell us, that if the price-level in such and such a year be denoted by
100 it is now 127 (or thereabouts) ; or that, if the cost of living of the
working classes in London be denoted by 100, in this or that provincial
town it is no more than 85 or 90. There are many different types of such
quantities and it is not easy to frame a short definition to cover them all
which shall be both precise and intelligible. In the majority of cases the
index-numbers are calculated over a series of months orjears and attention
is directed to their variation in time, but comparisons also fall to be made
in space, as in the case of cost of living in different towns above, or, to take
illustrations from other fields, if we wish to compare standardised birth*

S90

INDEX NUMBERS

591

rates in different countries or shipping freight-rates in different sea-routes.
From the elementary view-point, it is perhaps easiest to regard an index-
number as a measure of central tendency in a group of items ; and many
of the index-numbers in common use are nothing more than weighted
averages of relative numbers for the several component items of the
concept in question.

25.3 Table 25.1 shows, in column (2) the average annual price of English
wheat, as recorded in the Official Gazette, for the years 1930-1945 inclusive.
In column (3) we show these prices expressed as a percentage of the price
in 1930, and in column (4) the prices are similarly expressed as a percentage
of the price in 1945.

TABLE 25.1. — Prices of English wheat

(1)

Year

(2)

Price

(per quarter)

(3)

Column (2)
as percentage of

1930 pnce

(4)

Column (2) as
percentage of

1945 pnce

1930

s. d.

34 3

100

24 0

25 0

22 10

20 2

22 2

30 9

40 0

117

28 11

21 5

1940

42 10

125

62 10

183

102

68 6

200

111

69 8

203

113

63 11

187

103

61 10

181

100

The figures in columns (3) and (4) are very simple cases of index-numbers.
The eye cannot very easily follow the variations in price by running down
column (2), particularly if it is desired to gauge the magnitude of the
variation through time. By expressing the figures with reference to the
basic number of 100 we are, effectively, reducing the data to a convenient
common scale. Such figures are usually called “ price-relatives."

25.4 Simple as this example is, it brings out several points of practical
importance which are apt to be overlooked in dealing with the theoretical
problems arising from more complicated types of index numbers.

{a) Arithmetically the series of columns (2), (3) and (4) are equivalent
in the sense that they are proportional. Nevertheless, they may not
convey the same impression, particularly to the lay reader. It is very
natural to take the basic figure of 100, not as a purely convenient arith-
metical quantity, but as some " norm " or ** standard " of what ought

592

THEORY OF STATISTICS

to be. In our example, to say that the price in 1942 was twice as great
as in 1930 may convey a different impression from saying that the price
in 1930 was hdf that of 1942 ; in the first case we are taking the earlier
year as the standard of comparison, in the second case the later year. A
consumer of bread would probably incline to the former, an arable farmer
to the latter. This kind of point becomes of special importance for
economic index-numbers (such as those of wages or cost-of-living) which
are likely to be the subject of controversy. It must always be remembered
that the choice of the base-year may have to be exercised on grounds
other than those of convenience to the statistician, or those which might
appear to him of most importance.

(b) It is common practice to refer to changes in an index-number from
one year to another as a movement of so many points e.g. the index
in column (3) of Table 25.1 fell by 6 points between 1932 and 1933. There
is no great objection to this phraseology if the basic year is borne in
mind, but it is apt to provide a misleading picture of the importance of
the movement. The index also fell 6 points between 1944 and 1945 but
clearly the relative fall in the second case was smaller than in the first
(in fact, only about a third as great).

Price index-numbers

25.5 To fix the ideas, let us suppose that we require to construct an
index for the United Kingdom of wholesale prices over a series of years.
We shall first of all have to decide what commodities are to be covered by
the index and how to collect the prices. This leads to a number of practical
points which are apt to be troublesome (e.g. how to pick a representative
set of commodities, how to treat imported articles, and how to deal with
missing price-quotations) but which we shall pass over as not offering
any special theoretical problems. We will suppose that we have m
commodities whose prices in the ;th year are typified by p^j ^ • Pmr

These are heterogeneous quantities, each of them representing, it is true,
" money per quantity '' (for that is what^ is meant by a price) but the
quantity in terms of which the price is stated var 5 dng from commodity
to commodity. For pig-iron it is, say, a ton ; for raw cotton it is also a
weight, but the weight is only a pound, and for a precious metal only
an ounce ; for beer or wine it is not a weight at all but a volume ; for
woven textiles perhaps a “ piece and so on. In order to apply any of
the conceptions or methods of previous chapters, e.g. frequency distribu-
tions, averages, measures of dispersion, etc., we require a homogeneous
set of quantities all of the same dimensions ; as stated in 5.4 an average

is merely a certain value of the variable, and is therefore necessarily of
the same dimensions as the- variable so that if the data are of differing
dimensions the average has no assignable meaning. As an initial step,
therefore, we want to convert the heterogeneous figures of our price-table
into a homogeneous set of figures all of the same dimensions. This can
done in paore ways than one, but the simplest is to apply to each

INDEX NUMBERS

593

column of our prices-table the process used in Table 25.1, i.e. to convert
the given prices into price-relatives. As these are simple ratios, they are
all pure numbers. Table 25.2 illustrates the procedure ; Col. 2 repeats
the wheat prices of Table 25.1 and in Cols. 3 and 4 are added the Gazette-
prices of Barley and Oats, In Cols. 5, 6, and 7 these pnces are converted
into price-relatives with 1930 as base-year,

TABLE 25J2. — Prices and price-rdatlves of wheat, barley and oats

(1)

Year

Price per quarter

Pnce relative (1930=100)

(2)

Wheat

(3)

Barley

(4)

Oats

(5)

Wheat

(6)

Barley

(7)

Oats

1930

100

lOO

100

103

113

101

109

101

109

104

103

117

138

139

129

123

112

113

1940

125

229

217

183

303

238

165

200

586

245

1 69

113

203

398

254

187

335

264

181 1

316

267

25.6 In terms of our s 5 mibols then, we replace each price pf^ by a price-
relative trp where, ignoring the factor of 100,

.... (25.1)

prs being the price of commodity r in the standard year (or, to put it more
generally, the standard price of commodity r, for prices in a single year
are subject to casual disturbances and it may be better to take as standard
the average price over a five or ten year period). We may now average
the relatives (25,1) in any way we please in order to obtain our desired
index-number for the “ relative general level of prices If we take the
simple arithmetic mean of the f's, we have, using to denote this form
of index-number for the year j and S to denote summation for all com-
modities r

For instance, in the data of Table 25.2, ^Ii9so==100 and^Ju«=J{181 +
316+ 267) ==255.

594

THEORY OF STATISTICS

This formula, however, attaches precisely the same * weight ' to each
commodity whether little is- sold at the specified price or much ; a com-
modity such as wheat is given no more weight than a commodity such
as pepper, in spite of the enormous difference between the quantities
moving into consumption. We have therefore to consider whether some
system of weights can be introduced to allow for this effect.

25.7 Since our price-relatives are all of the same dimensions (pure
numbers) our weights should also be all of the same dimensions. They
cannot therefore be quantities, for some of the quantities are actual weights,
some volumes, and so on. Suppose then we make the weight for each
price-relative the momy spent on that particular commodity in the base
year (or the average annual amount in the base period), say prs where
is the quantity in question. Then for form B of the desired index-
number we have

T — ^ Prs ?ra)

(25.3)

S iPn g»)

^ {Prt ?rs)

(25.4)

This is a remarkable result, for (25.4) is simply the ratio of the cost of
the given “basket of goods'" (the quantities sold in the base year or on
an average in the base period) at the prices of year / to its cost at the prices
of the standard year or period. Looking at the matter in another way,
we have converted our heterogeneous price-figures, as required, into
homogeneous figures by multiplying each price by a quantity expressed
in the same units as are used in specifying the price and thus turning them
from “ money per quantity into “ money."

25.8 Although the index-number has a fairly intelligible meaning, it
is still open to some objections. In fact, it depends on the quantities sold
in the basic year, and if the actual quantities vary substantially from
year to year there is some ground for arguing that such a fact ought to
be taken into account. For example, if over a period the proportion of the
average household income spent on food drops from 40 per cent in the
basic year to 25 per cent, it seems obviously wrong to continue to weight
food-prices by a factor of 40 per cent. Our weights, so to speak, ought
in some sense to be kept up-to-date,

25.9 Before discussing this problem in generality, let us make four
preliminary observations —

(a) We noted in 14,15 that errors in weights, if uncorrelated with the
prices to which they are attached, will not exert much effect on the index
numbers. Thus, if the weights change rather erratically or by small
amounts from year to year, the accuracy of the index is not seriously
affected. For practical purposes, therefore, ^ should give a reasonable

INDEX NUMBERS

595

comparison between years which are not far apart in time and may be
satisfactory over quite a long period unless there is some systematic move-
ment in weights during that period.

(6) Purely practical difficulties in determining weights from year to
year may make some formula of the type (25.4) the only one which can be
calculated in time to be of any value.

(c) It is arguable on theoretical grounds that (or some similar
formula based on a different type of average) is the correct form to use
in estimating price changes. If we make allowance for changing quantities
we may be confusing price change with other things. For instance, an
index of the form !!(/>„ I'^iPrs ^rs) measures the ratio of the total
expenditure in the jth year to that in the basic year, and to that extent
is a definite measurable quantity. But when we try to dissect that part
of it which is dUe to price change from the part due to change in quantity,
we are in difficulties, for so far as obseivalion goes the two things are
really inextricable parts of the same phenomenon. There is, in fact, an
element of convention in our definition of a price index-number. The
statistician will always remember how his index-number is calculated and
will know how far he can use it in any particular argument. If he chooses
to define his price-index by reference to the fixed basket of goods he
is perfectly entitled to do so. He may perhaps be challenged on the
grounds that his price-index does not possess some desirable properties
which might be expected of a perfect price-index. He cannot fairly be
accused of doing anything wrong ; only of doing something inexpedient.

25.10 We have attempted to simplify the discussion by speaking of prices
and years in the construction of our index-numbers. Evidently similar
considerations apply when the periods of comparison are not years, but
some other unit of time, except that for short periods, such as months,
we may have to pay some attention to seasonal effects. Most of what we
are saying about price-indices also applies to other forms of index-numbers
although there are certain features of prices which give rise to special
difficulties. Broadly speaking, the theory of price-indices covers the
general case, and indeed other index-numbers are frequently much easier
to construct when they can be freed from measurement in terms of money*
We shall refer to the so-called quantum indices below (25*20). In the
meantime, we continue to discuss price indices on the understanding that
our discussion has a somewhat wider application.

Ceometxic means

25*11 The same kind of considerations which led us in Chapter 6 to
expiess a preference for the arithmetic mean in determining averages
also apply to its use for index numbers, except perhaps that the argument
from sampling simplicity is not so strong* The use of medians and modes
is to be deprecated and only the student of statistical history is likely
to encounter them in connection with index-numbers. There is, however,

596

THEORY OF STATISTICS

something to be said in favour of the geometric mean, particularly in
connection with price indices. Let us note the formulas corresponding to
J and

For the index based on the geometric mean of a set of prices relatives

= . . . . (25.5)

where m is the number of prices concerned. For purposes of calculation
this is more easily written as

log ,7, = ijs(logA,)-SaogA,)l . . (25.6)

ffl

Clearly (25.6) can also be written as

log ^S(l 0 g^„/A,) . . . (25.7)

It makes no difference whether we take the ratio of the geometric means
or the geometric mean of the ratios.

For the corresponding index to we have

f ( <lr$ j \ )

= j jp,, )

which is more conveniently written

log Jj

2 \qr,log{p„lpr,)

Example 25.1. — As an example of a price index-number calculated from
the arithmetic mean by reference to a fixed set of weights in a basic
period, we consider the British official ** interim index of retail prices."
This used to be known as the " cost-of-living index," a term which the
authorities are attempting to abandon in favour of a more neutral type
of wording. A better phrase would be " household budget price-index ",
since the object of the index is to measure changes in the average retail
prices of the items composing the expenditure in an average household
budget. The two main practical questions for decision in constructing
the index are ; what commodities are concerned and what is their relative
importance in the " average budget " ?

For the index-number, which was first published in 1947, the Ministry
of Labour used data collected in 1937 /9 by sampling about 10,000 house-
hold budgets. The information gave, in considerable detail, the expendi-
ture on all items for four separate weeks in October 1947, January 1938,
April 1938 and July 1938, and an arithmetic average of the four was
regarded as representative of the proportionate expenditure on each item
over the year. Some of the budgets were collected from agricultural
households and were separated for the construction of an index relating
to agricultural workers.

tKi)EX NUMBERS 59?

There are about 90 items involved and they are classified into eight
groups —

1. Food

2. Rent and rates

3. Clothing

4. Fuel and light

5. Household durable goods

6. Miscellaneous goods

7. Services

8. Drink and tobacco.

Current prices for the 90 items are collected by the Ministry of Labour
from various sources, e.g. by visits of local officers to retailers in regard
to food or by inquiries of local authorities and property owners* associations
in regard to rent. These prices are related to the corresponding figures
for the basic date, namely, 17th June 1947, taken as 100.

It then remains to compound these price relatives into an index for each
group, and finally, to compound the eight resultant indices into a single
index. The same principles are employed in each case and effectively
amount to the use of equation (25.3). They may be exemplified by the
method of constructing the final index from the eight component indices.

In calculating the final index, a weighted arithmetic mean is taken of
the components, the weights used being as follows

Food . , . .348

Rent and rates . . 88

Clothing . . .97

Fuel and light . . 65

Household durable goods 71
Miscellaneous goods . 35

Services . . .79

Dnnk and tobacco . .217

Total 1,000

Thus for instance, the index numbers of the eight groups in mid-December
1947 were respectively 103*4, 100*1, 102*4, 107*1, 106*3, 109*2, 102*5,
104*1. Taking our origin as 100 we have for the index for “ all items '*

100-f {(348 X 3 • 4) +(88 X 0 • 1) + etc.

+(217x4*1)} /1,000- 103*7

The weights in this case are an attempt to represent the proportional
expenditure in 1947 on the eight groups, e.g. it is estimated that 34*8 pei
cent of household expenditure was devoted to food. As no definite
information for 1947 was available, the proportions shown m the budget
inquiry of 1937/8 were adjusted to take account of changes in price

598

THEOKY OF STATISTICS

between 1937/8 and mid-June 1947. The proportion attributable to
drink and tobacco was scaled up to take account of 1947 conditions.

Example 25.2.— An illustration of a price-index calculated by the use
of geometric means with a fixed set of weights is provided by the British
index-number of wholesale prices. This Index purports to measure the
movement in the prices of wholesale commodities. It was revised in
1935 on the basis of information obtained from the Census of Pioduction
of 1930.

There are 200 commodities composing the Index, the numbers, in
eleven groups, being as follows —

Group

Number of
Commodities

Cereals ....

. 20

Meat, fish and eggs

. 20

Other food and tobacco

. 28

Total — Food and tobacco

Coal .....

. 9

Iron and steel

. 37

Non-ferrous metals

. 8

Cotton .....

. 10

Wool

. 11

Other textiles

. 9

Chemicals and oils ,

. 15

Miscellaneous.

. 33

Total-Industrial items

. 132

Total — All articles

. 200

These numbers, which are efectively weights for the groups concerned,
are based approximately on the relative importance of the various items
as indicated by the production figures in the 1930 census and imports
in that year, importance for this purpose being measured by the value
of the gross output.

Prices are obtained from various sources, mostly from trade publications,
and relate to certain standard types or specifications. In some cases the
prices of two or more qualities are averaged for a particular commodity
so as to give a wider coverage.

In the construction of the Index for any particular commodity the
price is recorded weekly where possible and an arithmetical average of
the weekly quotations provides a figure for the month. This is then
related to the price in the corresponding month of the basic year by means
of a simple price-relative, A composite index is then constructed for the

INDEX NUMBERS

599

month for each of the groups specified in the above table by taking a
geometric mean (the actual arithmetic process is somewhat different, but
this is what it amounts to).

The monthly index for “ All Articles is obtained as a geometric mean
of the price-relative for the 200 items in the above table. This is equivalent
to taking a weighted average of the eleven groups with weights given by
the number of commodities listed above. An annual index is constructed
by taking the geometric average of the index numbers for the twelve
months.

It will be noticed that in neither of the two examples we have just
given — two of the most important industrial indices in the United
Kingdom — are the weighting factors actual quantities. For the budgetary
index they are based on proportional expenditure in a standard period,
for -the wholesale price index they are based on value of gross output in
a standard period.

The time-reversal test

25.12 Let us now consider generally some of the properties which we
should like to have in an index number. We will not dwell on properties
such as ease of calculation, but will discuss some desiderata which arise
from our general notion of the functions which an index number ought
to perform.

In discussing the price-relative of Table 25.1, we noted that the series
of columns (3) and (4) were equivalent in the sense of being proportional.
The difference in the base year makes no difference to the index numbers
except one of scale. To put it slightly differently, the relative of year a
based on year 6, say kab, is the reciprocal of the index of year b based on
year a, say kba (except for the factor of 100 which we may ignore for
present purposes). That is to say kah = 1.

The price relative therefore obeys what we may call a time-reversal
test ; and this is clearly a property which we should welcome in any
index number, for then our comparison between two years does not
depend on which year we regard as the base year. That is, we should like
an index number to obey the relations

Iba = 1 .... (25.9)

Of the four indices we have considered earlier in the chapter only one
obeys the time reversal test, namely When we introduce weights
appropriate to a fixed base-year the time-reversal property is destroyed.
For instance, with ^ we have

^lah ~ S {Pm (}rh) /S {pfh Jffc)

jfha = S {pfb I ^ [pm

^ and equation (25.9) is not obeyed unless the are equal or proportional
to the qrh*s

6oo

THEORY OF STATISTICS

Nevertheless the test may be approximately obeyed if the changes in
weights from qra to qfh are small or if they are not highly correlated with
the prices. For let

= qrb + Bf where Br is “Small. Then

7 7 — ^ S {prh{qrb-\- Sr)}

^ ^ 2 {Prb qrb) ^ {pra{qfb-\^ Br)}

( y. {S.U 1^-^ w y (s.. B.) )

which is approximately

X {Pra qrb) S {prb{qrb‘^ 8r)}

S (^Prb qrb) S ^f)}

1 1 ^ [ 1 j- ^

S {prb qrb) S {pra ^rb)

^ ^ S {Pfb Sr) S f Sr)

S {Prb qrb) S {Pra ^rb)

(25.10)

As the quantities Sr are small the two terms on the right in (25.10) will
in general be small ; and even if they are moderately large the terms will
be small if S {prb Sr) and S {pra Sr) are small, i.e. if Sr is only slightly
correlated with pra and prb ; or if pra— prb is small.

Similarly for J[ we have

log ji^ob jylba

S {qrb)

1 .
S(gn>)

log {pra! prb)
[qra log {prb j pra)

We may suppose that S (^r.) =S for the total "weight” may con-
ventionally be kept constant, and thus we find, after a little reduction,

log jjab iJba

h (qrb)

2 •! log {pra I prb)

This is nearly zero if the S's are small or if S is only slightly correlated with
the logarithm of the price changes and again the time-reversal test is
obeyed approximately.

25.13 In order to obtain an index number which is Certain to obey the
time-reversal test we may proceed as follows :

With the base year b, the index for a year a is given by

ra qrb)

® ** lj{prb qrb)

With the weights of the year a, but still with b as base, we have an index
number

^{P ra

INDEX NUMBERS

6oi

We then define

slob =

ra (][rb) S {Pra ^ra)
^ H {Prb ^rb) S {prb ^ra)

(25.11)

This was called by Irving Fisher the ideal index-number. He regarded
it as the best possible.

Examination of (25.11) will show that the time-reversal test is obeyed,
for the reciprocal of jgl is the product of o^ba and J'ba. The principal
difficulties in using the ‘‘ ideal number are practical ones ; we rarely
have data in sufficient detail to allow us to calculate it over a series of
years.

The factor-reversal test

25.14 Irving Fisher {The Making of Index-Numbers, 1922) also proposed
what he called a factor-reversal test for price index-numbers. He argued
that if we interchange the symbols for price and quantity we should reach
an index of quantity changes which, when multiplied by the index of
price changes, should measure the change in total value. Consider, for
instance, a^ab- For the price index-number we have

S iPrb qrb)

Now if we interchange p and q we have an index which we may write

S (^ra

S (^f& pr}^

(25.12)

This may be regarded as an index of quantity of type ^ weighted according
to the prices ptb in the basic year 6. Now we have

ra qrb) S {prb qrc^

But this is not equal to the index of total expenditure qra)l'E{prb q^b)
and hence the factor-reversal test is not obeyed.

25.15 Of the indices we have considered in this chapter only the ideal "
index obeys the factor-reversal test. The reader can easily verify that
this is so from equation (25.11). This was, to Fisher, a powerful reason
in favour of the “ ideal index. It does not appear to us to carry quite
so much weight as he attributed to it. There is an element of convention
in the construction of an index of quantity such as /, just as in the price
index itself and obedience to the factor-reversal test would appear to be
most required when indices of price and quantum (25.19 below) are required
together.

602

THEORY OF STATISTICS

The circular test

25.16 If an index is constructed for year a on base-year i, and for year
h on base-year c, we may derive an index for a on base-year c. The
so-called circular test requires that if we do so we ought to get the
same result as if we calculated direct an index for a on base-year c without
going through b as an intermediary. To put it another way, we require
that

lab Ibc Ica^l . . . . (25.13)

which presents a kind of extension of the time-reversal test of equation
(25.9). We may note in passing that we shall not require to examine
more complicated criteria such as

lab Ibc led Ida == 1 . . . . (25.14)

for such are always fulfilled if (25.9) and (25.13) are satisfied. For then
lal Ibc = 1 I lea, led Ida == 1 I lac

and hence the left-hand side of (25.13) becomes 1 / Ica lac = I

25.17 The circular test is obeyed by but not by any of the other
indices we have considered. Fisher, in fact, for reasons which we do not
regard as very cogent, argued that an index-number should not obey
the circular test. We need not dwell on the point, since it may be shown,
as for the time-reversal test, that the circular test is approximately obeyed
if weights do not change very substantially over the period for which
comparisons are being made.

Departures from the fulfilment of the circular test are perhaps more
important in comparisons in space than in time, for them the weights are
likely to differ to a greater extent. For example, index-numbeis pur-
porting to compare industrial production, cost-of-Hving or price levels
between different countries may depart from the “ circular ” criterion very
considerably. By the use of an appropriate set of weights it may be
possible to compare country A with country B ; but to compare either
with C new weights may be required. Hence it is quite possible to find
that the “ production for example, in A is greater than in B, and in
B is greater than in C , whereas a direct comparison can show that

production ” is less than * C*s. This inconsistency really implies that
we are trying to do too much with our index numbers. There are limits
to the amount of information we can compress into single numbers for
comparing areas or periods in which conditions are very different. The
most w’orkable method of approach is probably the one we noticed in
dealing with death-rates (14,17) where a standard set of weights is used
for each index.

INDEX NUMBERS

603

Example 25.3, — Moving weights

An interesting attempt to deal with the question of changing weights
was made in the official index of agiicultural prices introduced by the
British Ministry of Agriculture and Fisheries in 1938 (Houghton, J, Roy.
Stai. Soc., 1938, 101, 275). Between the two world wars the pattern of
agricultural production changed considerably in the United Kingdom
owing to the movement from arable to grassland farming and the introduc-
tion of some new crops such as sugar beet. Weights were calculated for
each year for the various items entering into the index, based on the
proportionate contribution by value to the total output. A five-yearly
moving average was taken of these weights and the weighting factors
used for any particular year was the value of this average for the five
previous years. In an industry such as agriculture, wherein changes
from yeai to year are not very large, this slow and continuous adjustment
of weights to current conditions has much to recommend it. Com-
parisons between years which are fairly close together can be made with
confidence.

Linking methods

25,18 Situations sometimes arise in which we may compare each of a
series of years with the next, but cannot so easily compare years which
are separated in time. This is particularly so when weights are changing
rapidly or when new commodities enter the market or disappear from it.
In such circumstances it may be possible to construct an index for year
2 based on year 1, for year 3 on year 2 and so on, and hence to construct
a continuous series by linking successive years. If, for instance, the index
for year 2 on year 1 is i^ and that for year 3 on year 2 is ^’3, etc., we may,
taking year 1 as base, regard i-J^ as the index for year 3, iJJ 3 as the index
for year 4 and so on. Comparisons for successive years are not invalidated
though those for widely separated years may be very unreliable. Index-
numbers of this kind are sometimes useful as presenting a general picture
of movements over a period ; but they are obviously not so firmly
founded from the theoretical viewpoint as those we have described
above-

Example 25,4. — Index number of shipping freight rates (Isserlis, J. Roy,
Siat. Soc„ 1938, 101, 53.)

It was desired to construct an Annual Index representing the course
of Tramp Shipping Freights over the period 1869 (when the Suez Capal
was opened) and 1936 when the calculations were carried out. From the
outset it is dear than any index of this character will require careful
interpretation, for the period concerned was one in which sea transport
was revolutionized by the change from sail to steam, and later a further
partial change to propulsion by Diesel engines* Furthermore, details of
the freights for all voyages undertaken in this period were not available,

6o4

THEORY OF STATISTICS

and the actual quantities carried were also not available. In spite of the
unpromising conditions of the problem an index was constructed on the
following lines —

Quotations of the highest and lowest freights in a particular year were
available over the period concerned and the mid-pomt between the two
was taken as representing the average freight for the year. This is a crude
form of average necessitated by the paucity of the data, but is probably
reasonably accurate except in years such as 1915 when freights trebled
as compared with the year before owing to the circumstances of World
War L

These average freights were available for 210 homeward routes to the
U.K. and 112 outward routes, but owing to the varying nature of the
tramp trade over the period, quotations were not available m respect
of each route for each year. Consequently for any particular year there
were a number of missing quotations. For each route where quotations
in consecutive years were available a price-relative was constructed
based on the previous year; for example, for the route Java /U.K. in
Sugar the freight in 1870 was 93 per cent of that in 1869 and the price-
relative was therefore 93. In 1919 the freight was 31 per cent of that for
1918, and the price-relative was therefore 31.

For each year the available price-relatives were averaged arithmetically
over homeward and outward routes to give an average price-relative for
that year as compared with the previous year. For example, the average
price relative for 1936 was 117*3.

An index over the 68 years concerned was then constructed on the
basis of a chain method. The average pnce-relative for 1870 was 103,
and on the basis of 1869=100 the freight index was also 103. The average
price-relative for 1871 was 99 and the index was therefore taken as
(99x 103)/100, namely 102. Similarly, by this chain method, the index
was built up from one year to the next. The freight index for 1935
was 88. The price relative for 1936, as noted above, was 117*3 and
therefore the index for 1936 was (88x117*3) /lOO, namely 103.

For the purpose of giving a general view over the period, the index
is perhaps not unsatisfactory. Although the rates on individual routes
cannot be weighted by reference to the quantity of traffic the large
number of routes employed ensures some degree of weighting in the index
as a whole according to volume of traffic ; and although a comparison
between neighbouring years is more reliable than one between two years
which are widely separated in time, it is between the closer years that
comparisons most frequently fall to be made.

In 1935 there became available detailed information of tramp. voyages
undertaken in U.K. ships in that year.. It was then possible to construct
an index of tramp shipping freights weighted according to gross freights
earned on cargoes carried in that year. The agreement of this index
with the chain index was fairly good.

INDEX NUMBERS

605

Quantum indices

25.19 Reduction of the data to homogeneity is a pre-requisite of all
index numbers of the type we have considered in this chapter, and we have
already noticed that in many instances the only available common unit
is money value. Unfortunately, this is precisely the unit which does
not remain constant over periods of time. If we measure the industrial
production of a country by the value of the gross output of its manu-
facturing plant and find that the value in 1948 was twice as great as in
1938 we obviously gain a very poor idea of the change in output over the
period in any real sense. Our prices have changed in the meantime.
Can we then measure in any reasonable way what is the change in output
apart from changes in prices ? Can we obtain some index of production
which is related to physical output and is free from changes in prices or
money values ?

25.20 Suppose that in the basic period the value of the output of a
commodity is typified by Vrs and the price of some unit by prs. If the
price in the;th year is prj, and the output is valued at Vrj, then the quantity
Vrj pfs I prj is what the output would have been valued at if the price had
been that of the basic year. We may then construct the index-number

S (Vfj pniprj)
S {Vrs)

(25.15)

This is the ratio of the value of the output in the jth year, revalued at
basic prices, to the value of the output in the basic year. It evidently
goes a long way to meet our requirements. It bears a kind of inverted
relation to the index of equation (25.4). If there exist quantities q such
that Vr}=pr 3 qrj wc havc, on substitution for v in (25.15)

S {p,s q„)
' ^ " S (Prs qrs)

(25.16)

which exhibits as an average of quantities q weighted by prices in the
basic period — a similar index to that of equation (25.12). As noted in
25.15, the factor-reversal test requires that our indices of price and output
shall, when multiplied together, measure the change in value of total
output — a very reasonable requirement when both indices are used but
not necessarily a desideratum when only one of them is to be calculated.

25.21 It is of some importance to note that we can calculate from
(35.15) even when quantities q do not exist. Suppose, for example, we
are constructing an index of the price of travel in London, into which
there enter expenditures on buses, trams, electric trains and taxis. There
is no “ quantity '' of travel though perhaps we might construct measures
on a mileage basis. This, however, is unimportant if we know the ex-
penditures V and the ratios prjfprs; if, for instance, we know that in the

6o6

THEORY OF STATISTICS

yth year prices of fares on buses, trams and trains are 10 per cent greater
than in the basic year, whereas taxi-fares have remained unchanged. So
long as the price-relatives are known, the expenditures % and Vrs are
sufficient for the computation of without the intermediate calculation
of notionary quantities q.

Index-numbers such as pi are best known as quantum indices. Ex-
pressions such as " index of volume occur but are misleading as the
following example shows.

Example 25.5. — The British Board of Trade publishes an index-number
of the “ volume of imports and exports. This is obtained by revaluing
imports or exports in the given period on the basis of 1938 prices and
expressing the results as percentages of the 1938 values. The following
are the figures for 1946 and 1949 (1938 == 100) —

1946

1949

Imports ....

Exports (including coal)

151

Exports (excluding coal)

107

161

Now it so happens in this ^'ase that we can estimate the actual weights
(in tons of cargo) covered by these import and export figures. For exports
(excluding coal) it is estimated that the figures were, in 1946, 98 per cent
of 1938 and, in 1949, 120 per cent. Thus, where the quantum index gives
161, the index based on actual weight in tons is only 120. Clearly the
quantum index does not measure volume in any ordinary sense
associated with physical size or weight alone (it may be regarded as an
index of this weight weighted by prices). On the other hand, it may be
the correct index to use when attention is being directed to the relative
contribution of exports to the balance of trade, price changes being
eliminated from the comparison with the basic year.

25.22 In conclusion, we may intimate, without being able to pursue the
subject, that for certain classes of statistical work it appears to be possible
to develop a theory of index-numbers of a rather different kind from that
discussed in this chapter. Psychologists have for some time studied
techniques for isolating " general factors from a complex of tests of
ability which are capable of application to the isolation of a general price
level ** from a complex of price movements. Biometricians, from a
different point of view, have considered the problem of forming linear
functions of observations which will most closely, in some reasdnable sense,
summarise the essential properties of classes — the so-called discriminant
functions Something has already been done in applying such methods
to the formation of index-numbers. The subject has, however, hardly
reached the point of practical application in economics, and it is unlikely
that the methods described in this chapter will be supplanted for general

INDEX NUMBERS

607

SUMMARY

1. The price-relative of a commodity for a particular period is the
ratio of its price in that period to the price in a basic period. It is usually
multiplied by 100 for convenience of expression as a percentage.

2. There is an element of convention in the definition of a price index-
number. Simple unweighted numbers are

aI} ” iPnlpr^)
c/, = I n iPrilPn) p

3. Weighted index-numbers in common use are

= S {pf} qrs) I S {prs ?rs)

log jjJ) = 2 ^ I P^^) I

4. The time reversal test requires that

lab Iba ^ I

This is obeyed by qIj but not by the weighted indices, though the latter
may obey it approximately.

5. The time-reversal test is obeyed by the ideal index-number

This also obeys a factor-reversal test.

6. The circular test requires that

lab Ibc Im = 1

It is not obeyed by any of the weighted indices unles^ the weights are
constant, but may be obeyed approximately.

7. Linking methods may give a suitable chain index when data are
available to make comparisons possible for adjacent years.

8. Quantum index-numbers purport to measure a "quantity'* in-
dependently of price change. The principal form in common use is

^ = S {Vfj Prs I prj) I S {Vfs)

Quantum does not necessarily measure physical volume or weight.

6o8

THEORY OF STATISTICS

EXERCISES

25.1 The following figures show the wholesale prices of refined petroleum
per gallon in the U.K. for the years specified. On the basis of 1923=100
construct a series of price-relatives.

Year

1923

8
9

1930

Pnce per gallon
(pence)

12 |

i2i

llj

10 |

lOi

25.2 Show that the index-number possess the “ chain-property ” of
25.18, namely that the index for a year j on base 1 is the product of
corresponding indices of y on 1, y— 1 on j—2, . . . , 2 on 1.

25.3 The following figures show for U.K. total imports (a) the declared
value and (b) the value on the basis of average values in 1930. Taking
1930 as a base year construct index-numbers (1) of average values and

(2) of quantum for the years 1931-~6.

Year

Declared Value

Value on 1930 basis

^ million

£ million

1930

1,044

861

1,067

702

939

675

946

731

991

756

1,012

848

1,077

25.4 Using the weights of Example 25.1 calculate the index for all
articles if the indices for the constituent groups are as follows : Food
95 ; Rent and rates 90 ; Clothing 110 ; Fuel and light 120 ; Household
goods 102 ; Miscellaneous 115 ; Services 98 ; Drink and Tobacco 108.

Examine the effect of rounding up the weights (a) to the nearest 10
(J) to the nearest 100.

INDEX NUMBERS

609

25.5 In the notation of 25.14 consider the index-number

+ </&«)

Show that if the weights in the years a and h differ by a snjall amount
the difference between this index and the ideal index, is zero to the
first order in dr,

25.6 The following figures give the annual average prices in the U.K.
for beef, mutton and pork.

Year

Beef (pnme)

Mutton (pnme)
pence per 8 lb.

Pork

1935

1940

101

102

Construct an index of meat prices for the period {a) of type J, (b) of
type gl with weights 4, 2 and 1 for beef, mutton and pork respectively.
Take 1935=100 in each case.

25.7 Show that the index-number

T — ^

^ H {prb{qra’i-qrb)}

obeys the time-reversal test but not the circular test unless the w'eights
in the three years a, b, c are equal.

CHAPTER TWENTY-SIX

TIME-SERIES

Introduction

26.1 When we observe numerical features of an individual or a population
at different points of time, the set of observations constitutes a time-senes.
The temperatuie at a given place over a given period, the population of
a country over a number of years, the imports of a country for a series
of months, the weight of an animal recorded at various stages of growth,
are familiar examples of the kind of phenomena which provide series of
values at a succession of points of time. The statistical data which
they furnish differ from most of the data which we have discussed hitherto
in that we are interested, not merely in the aggregate of values, but in the
order in which they occur.

26.2 Throughout this and the succeeding chapter we shall consider only
series of values given at equidistant intervals of time. By taking the
time-interval as unit we can then regard our series as defined at times

2, 3, etc., and can write the values of the series as etc.,

the value at time i being u^. If for any reason we wish to reckon time
backwards as well as forwards from time ^=0 we can write the series as
w-i, Uq, etc.

The restriction as to equidistant intervals is not in practice a serious
limitation. Most series which are available in official publications such
as economic, demographic, and meteorological series, are in fact given
at intervals which are exactly equal, as days, or approximately equal, as
years, or more or less roughly equal, as calendar months. Experimental
data are usually collected at equidistant intervals as a matter of routine
or are recorded (as on barometric graphs) in a continuous form from which
equidistant readings may be taken. Our discussion of theoretical questions
is greatly simplified by assuming equidistance in the time-intervals.

26.3 Although we shall draw all our illustrations from time-series it
should be pointed out that the theory is also capable of application to
certain other types of statistical data. For instance, if we put a thread
of cotton under the microscope it presents, as we proceed along the thread,
a fluctuating profile which bears at least a superficial resemblance to an
oscillating time-series ; and we can regard the nitrogen content at various
points along a strip of soil as the values of a series in which the time

6io

TIME -SERIES 6ll

variable is replaced by a space variable. In fact our methods are
applicable, and are often appropriate, whenever we have a statistical
variable depending on a variable t, whether relating to time or to linear
space.

26.4 In general the variable u may be discontinuous or continuous,
univariate or multivariate. For example, numbers of human beings are
necessarily integral and population-series are therefore discontinuous in
the variate ; on the other hand rainfall and temperature are continuous.
Again, we may wish to study the movement through time of one variate,
such as the price of wheat, or of several, such as wages, employment and
volume of industrial output. In the latter case it is usually more con-
venient to regard each variate as yielding a separate (univariate) series
and to study the relations between variates as the joint variation of
several series.

26.5 Although our time-values are discontinuous, we must remember
that the series itself, of which they form equidistantly spaced observa-
tions, may be either continuous or discontinuous in time. Some series
are necessarily discontinuous. For example, the final dividends on an
industrial security are declared once a year, usually but not always on
about the same date, and there are no variate values between those dates.
Again, although the act of earning an income may be carried on almost
continuously, the remuneration received is usually paid once a week,
once a month or once a quarter, namely at discontinuous intervals. Some
series are continuous and may be continuously recorded, as for instance,
by the instruments which graph on a rotating drum the temperature and
barometric pressure in a particular locality. Between the extremes of
unambiguous discontinuity and continuity we find numerous cases of a
hybrid character. The price of a loaf of bread may be regarded as
existing continuously while shops are open and even perhaps while they
are shut ; the price of an industrial share can hardly be regarded as
existing while the Stock Exchange is closed, and when it is open really
varies discontinuously in the sense that on an active market the price
may change with each transaction and hence is only determined at
particular moments during the day. Certain quantities such as annual
income or monthly -rainfall are discontinuous in the time- variable in so
far as there is only one value for the year or month as the case may be,
but continuous in the sense that they are an accumulation over a con-
tinuous period of time. Such distinctions will not often cause us difficulty
but they provide one more illustration of the maxim, of which perhaps
the reader may be growing a little weary by this stage, that one should
never forget the nature of one's primary material.

Some examples of time-series

26.6 We now give a few illustrations of the kind of material which we
have to study in practice. Some examples have occurred earlier in this

6i^

theory oe statistics

book. Table 15.6 (Fig. 15.6) on page 359, showing the population of
England and Wales at ten-yearly intervals, gives a typical series for the
growth of a large aggregate of human beings. The series is smooth in the
sense that the values he closely about a continuous curve. On the other
hand, the infantile and general mortality rates of England and Wales
graphed in Figure 13.1 on page 318, though moving downwards over the
period covered by the diagram, do not decline regularly. Table 26.1
and Figure 26.1, showing the sheep population of England and Wales
for certain years, give a picture of a somewhat similar kind, but the
departures from a smooth movement are of longer duration, and it is not
easy to decide from these data whether the increases following the low
point in 1922 are a reversal of the downward movement or only a tem-
porary fluctuation.

TABLE 26.1. — Sheep population of England and Wales for each year from 1867 to 1939

Data from the Agricultural Stattshcs

Year

Population

(10,000)

Year

Population

(10,000)

Year

Population

(10,000)

Year

Population

(10,000)

1867

2203

1886

1892

1905

1823

1924

1484

2360

1919

1843

1597

2254

1853

1880

1686

2165

1868

1968

1707

2024

1991

2029

1640

2078

2111

1996

1611

2214

2119

1933

1632

2292

1991

1805

1775

2207

1859

1713

1850

2119

95 1

1856

i 1726

1809

2119

1924

1752

1653

2137

1892

1795

1648

2132

1916

1717

1665

1955

1968

1648

1627

1785

1900

1928

1512

1791

1747

1898

1338

1797

1818

1850

1383

1909

1841

1344

1958

1824

1384

26.7 The two last examples exhibit not only local variation but a broad
movement over the period, a trend as we may call it. In our next three
examples there is no apparent trend but varying degrees of ‘‘ short-term ''
or local variation. Table 26.2 and Figure 26.2 show the percentage
losses of British ships per annum (i.e. 100 times the tonnage lost divided
by the tonnage at risk)*. There is a good deal of variation from year .to
year but it is not very regular, at least so far as the eye can judge. In
Table 26.3 (Figure 26.3) showing the crude birth-rates of cattle in Great
Britain on a quarterly basis there is, in contrast, a marked regularity due

TIME - SERIES

613

to the seasonal character of births of cattle. There may, of course, have
been seasonal effects in the data of Table 26.2, but if so they have been
obliterated by the use of annual figures. Table 26.4 and Figure 26.4
show a rhythm^in numbers of sunspots which is not seasonal. It is not
so regular as that of Table 26.3 but there is evidently some degree of
regularity present.

1865 1885 1605 1925 194S

Years .

Fig 26.1~-Graph of the data of Table 26.1

TABLE 26.2.— -U.K. vessels lost as a percentage of the total U.K. fleet in certain yean
Vessels of 100 g.r.t, and over

Figures from Lloyds Register StattsttceU Summary

Year

% loss

Year

% loss

1920

0*68

1930

0*49

0*34

0*17

0*62

0*30

0*73

0*39

0*57

0*42

0*32

0*41

0-58

0*24

0*34

7 '

0*46

0*59

8 I

0*50

0*56

6i4

THEORY OF STATISTICS

Fig. 26.2. — Graph of the data of Table 26.2

26.8 Examples such as these lead us to regard a time-series as composed
of three constituent items, a long-term movement or trend, a short-term
systematic movement and an unsystematic or random component. Some
series, of course, do not exhibit all three — the movement shown in Figure
15.6 is nearly all trend, that of Fig 26.3 is nearly all systematic oscillation,
and that of Figure 26.2 seems on the face of it to contain a good deal of
random fluctuation. One of our principal problems is to isolate these
components for separate study.

TABLE 26.3.— Crude birth rates (number of births per 100 population) of cattle in

Great Britain

Data from Joan Marley, /, Roy. Stcd. Soc., 110., 187

The figures have been multiplied by a factor of approximately four to make them
comparable with annual rates

Year

December-

February

Birth rate

March-

May

June-

August

September-

November

1940

33 2

45-2

33*2

40*0

35-2

44-0

38*8

32*8

35-2

46*4

35*6

34*4

34*8

44 ‘8

32*0

38*4

37^6

41*2

82-8

30*8

36-0

42*0

30*0

35*2

Crude birthrate

TIME -SERIES

615

Year

Fig. 26.3. — Graph of the data of Table 26.3

TABLE 26.4. — Wolfer’s sunspot numbers for the years 1853-1900

Quoted by G. Udnv Yule, Phil Trans. 226, 267

Year

Number

Year

Number

Year

Number

Year

Number

1853

39-0

1865

30-5

1877

12*3

1889

6*3

20-6

16*3

3*4

1890

7*1

6-7

7*3 -

6*0

35*6

4*3

37-3

1880

32*3

73*0

22-8

73*9

54*3

84*9

54-8

1870 '

139*1

59*7

78*0

93*8

111*2

63*7

64-0

1860

95-7 i

101*7

63*5

41-8

77*2 '

66*3

52*2

26*2

59*1 1

44*7

25*4

8 I

26*7

44*0

17*1

13*1

12*1

.4 !

47-0

11*3

6*8

1900

9*5

26.9 One initial word of warning is necessary. It is useful to isolate
the components of a senes for sundry purposes. We may, for instance
be interested in the broad movement of a series and hence concentrate
attention on the trend to the exclusion of local and casual variation. But
this does not necessarily mean that we can in a parallel manner isolate
the causal systems underlying these movements. As a pure matter of
description we may ignore local variations and consider the trend ; but we
must not mislead ourselves by supposing that there is some fundamental
cause or set of causes which generates the trend movement and another
distinct set which accounts for the local movements. This is sometimes

6i6

THEORY OF STATISTICS

SO, but not always so. We shall give later in the chapter (Example 26.4)
an example of an artificial series which reproduces most of the features
of the so-called trade cycle, namely a series of long swings on which are
superposed more erratic short-term movements, but which is not composed
of a trend-generator and a short-term generator.

Fig. 26.4. — Graph of the data of Table 26.4

26.10 We may also remark at this stage that the distinction between a
long-term and a short-term movement is to some extent arbitrary. The
so-called trade-cycle is a long movement for most business purposes, the
depressions and peaks occurring about once every ten years on the average.
But in considering the recurrence of ice-ages or the growth and decay
of civilisations, ten years would be a very short time. What we call a
trend in any particular case is a matter of choice. It would be more
accurate to speak of long-term or short-term movements and even then
it is a convention what length of time we regard as long or short.

Trend

26.11 The general notion of trend as a broad continuous motion of the
system leads us to consider the possibility of representing it by a poly-
nomial in the time-variable t. The representation of a set of values

. . . % by a parabola of the form

(26.1)

has already been considered in Chapter 15 and we need add little to what
was said there on the subject. In Example 15.5 we did, in fact, fit
cubic parabola to the population data of Table 15.6 and obtained a very
fair fit.

TIME -SERIES 617

26.12 This method of trend determination has some serious drawbacks
when, as in Table 26.1, the polynomial required to obtain a good fit is
of high order. The arithmetic becomes troublesome ; the higher order
terms of the polynomial tend, as we pointed out in 15.22, to “ wag the
tail ” of the curve ; and if at some stage we add further terms to the
series (as frequently happens when new data arise by the passage of time)
the work of fitting has to begin afresh. The object of polynomial fitting
can be attained by a simpler process known as the method of moving
averages.

Moving averages

26.13 Consider the first 2m -{-I terms of the series, where w is a number
which we can choose at will. We may fit a polynomial of order ^ to these
terms and by convention will take our origin at the (?^i-|-l)th term, i.e,,
the middle one. Our polynomial, fitted by the usual method of least
squares given in chapter 15, will then be of the type (26.1) and we may
determine the constants aQ , , , a^hy such equations as

S(«iff)~aoS(/f)-«-fliS(i!^+i) . . . -a3,S(z:^+^)=0 . . (26.2)

there being {p + l) of these equations corresponding to values of / from
0 to p, and the summations extending over the values of t from ~ m
to + m. (Compare equations (15.8) on page 344 )

This polynomial is the best fit, in a least squares sense, to the first
(2^w + l) terms of the series and we may therefore take it as determining
the trend value at the origin, that is to say at the (m4'l)th point. The
trend value is then obtained by putting ^=0 in (26.1) and reduces simply to
We need therefore only determine from the equations (26.2).
The other constants a are not required.

It should be noted that the sums occurring in (26.2) are simply sums
of the integers or their powers from — w to m and hence depend
only on m and p, not on the values of Ui, except in the case of the first
term iff). It then follows that when we solve the equations for
we shall obtain a linear expression in the values Ui of the type

^0 ~ » • (26.3)

where the h*s depend only on m and p. This expression is merely a
weighted average of the first {2m +1) values of the series, the weights b
being determinate once we have fixed m and p.

We may now repeat the process by moving along the series and fitting
a curve to the (2m +1) points from to determining a trend

value corresponding to the point middle one of this set) ; and

since our treatment remains the same except for changes in the values
of the the trend value will be given by

where the 6's axe the same quantities as were reached in equation (26.3).

6i8

THEORY OF STATISTICS

We then proceed one step further along the series and repeat the process ;
and so on.

26.14 The net result of this treatment is that once we have determined
the constants b we can ascertain trend values by a weighted average of
sets of (2m-+-l) consecutive terms. We take in fact, a moving average
along the series. There will be no values corresponding to the first m
or the last m terms of the series and we must either resign ourselves to
having no trend for these 2m terms or adopt special measures to obtain
them. Our trend values will “ smooth the series in the sense that they
correspond to values of best fit given by polynomials of local application.
The process of trend determination is often described as “ smoothing.”

26.15 Let us consider the simplest case when we fit straight lines to
sets of three points, (w=-l, f = Our polynomial is then simply
and we have to minimise the sum of squares

which leads to the equations

=0 j

=0 I'

Now £(if)=0 and in general S(/^)=0 whenever p is odd.
simply from the first equation of (26.4)

(26.4)

We then have

ao == iE(Wf)

^ ’ (26.5)

In short, our trend value at any point is simply the arithmetic moan of the
three values of u centred at that point.

26.16 Consider next the case when we fit straight lines (^=1) to sets of
2m +l points. Corresponding to the first equation of (26.4) we shall have

2(i/e)— (2m+-l)ao =0

leading to-

^0 == • • • +^m) • • (26.6)

In simple generalisation of the previous case we then have the result
that the trend value at any point is the arithrnetic mean of the (2w+l) ^
Yalpes centred at that point,

TIME -SERIES 619

26.17 The next case in order of complexity is the fitting of a quadratic
parabola to sets of 5 points w=2). We then have to minimise

<«_2

and remembering that S(^)=0 for odd p we arrive at the equations

— 5^0 == 0

(26.7)

Now and i;(^^)=34. The relevant equations are then

li(Ui) — 5«o-~ 10^2 = 0
10ao“'34a2 = 0

leading to

=l|l7EK)-52(<%,) .

=— — 32^2! •

(26.8)

26.18 Proceeding in this way, we can determine the weights appropriate
to any system of m and p. The values of the weights for the cases required
in practice, however, have been worked out and the simpler ones are
given below. Let us note two properties of any system of weighting
given by this method —

{a) The sum of the weights is unity. This follows from the fact that
the sum in such an equation as (26 8) is obtained by putting all the u's
equal to unity. If we do this in the first equation of (26.7) and equate
all the other a's of even order to zero (as we may, since in this case a straight
line gives a perfect fit) we see that

{b) The weights are symmetrical about their middle value. This follows
from the fact that we must obtain the same result if we start from the end
of the series and work backwards.

We can then write a series of weights such as those of equation (26.8)

in the form ^ [—3, 12, 17, . . . ]. Those of (26.5) would similarly be
35

written - [1, 1, . . . ]. With this notation we can now write down,
3

without proof, the weights for the simpler cases.

620

THEORY OF STATISTICS

p^\ (straight line) —

2w+l

= 2 or 3 (quadratic or cubic) —
Values of ni

[ 1 , 1 , . . . 1 , . . ]

231

429

[-3, 12, 17 ]

[-2, 3, 6, 7, . . .]

[--21, 14, 39, 54, 59 . . .]
[—36, 9, 44, 69, 84, 89, . .]

= 4 or 5 (quartic or quintic)-
Values of m

231

429

[5, ^30, 75, 131, . . .]

[15, -55, 30, 135, 179, . . .]
[18, -45, -10, 60, 120, 143, . . .]

(26.9)

y. . (26.10)

(26.11)

The reader will note that the same formulse are obtained for ^=2A+1 as
for p=2k We leave it as an exercise for him to examine why this is so.

26.19 It is evident that expressions such as this rapidly become rather
cumbrous. We shall consider below how they may be simplified by
approximation, but before doing so will give a numerical example.

Example 26.1

To fit a trend line by moving averages to the sheep population data
of Table 26.1.

Let us first take a simple average of the type (26.9). We have to decide
on the extent of the average, namely the number m. Our process will
be sufficiently clear if we fit a curve to the first forty terms of the series
only.

There is, at this stage, no golden rule which can be laid down for the
determination of the extent of the average. We can only try a few values

TIME-SERIES

621

and see if they give us the kind of trend line we want. Let us then take
two values, m=2 and m=4 (corresponding to extents of 5 and 9 terms
respectively).

For the moving average of 5 we have to sum consecutive sets of five
terms and divide by 5. The process is illustrated m Table 26.5. It is
very simply earned out because in moving on a step we have only to add
on one term to the sum of five at the end and take off one at the beginning.
A similar process gives us the moving average of nine terms. Figure
26.5 shows the result of fitting the two trend lines.

TABLE 26.5. — Illustration of the arithmetic of fitting a simple moving average of fives
to the data of Table 26.1

(1)

Number of
term, t

(2)

Value of
senes

(3)

Sum of

consecutive sets
of five values
of Ut

(4)

i of column

(3)

(5)

Deviation,
column (2)
less column
(4)

2203

2360

3 i

2254

11006

2201

2165

10881

2176

- 11

2024

10735

2147

-123

2078

10773

2155

- 77

2214

10815

2163

•51

2292

109-10

2182

2207

...

. . ,

2119

• • •

. . .

Now let us try fitting a quadratic to consecutive sets of 7 points. The
appropriate formula is, from (26.10)

i [-2, 3, 6, 7. ... ]

This is not nearly so easy to apply as in our first case. We shall have,
for the initial term corresponding to />=3

1 {( -2 X 2203) +(3 X 2360) +(6 X 2254) +(7 X 2165)

+(6 X 2024) +(3 X 2078) -(2 X 2214)} = 2157

and a new calculation of this kind has to be done for each term of the
trend line. The process is straightforward but tedious. It may be
facilitated by the construction of a template which leaves only seven
consecutive terms exposed to view, so that the eye does not pick out the
wrong terms in machine calculations.

622

THEORY OF STATISTICS

Fig, 26,5

We have shown in Figure 26.5, the result of applying this process to the
series of Table 26.1.

An examination of this diagram will reveal the conventional nature
of the determination of trend. The 7-point quadratic is not, for most
purposes, a good trend line because it follows the primary data too closely
and reproduces short term fluctuations. The fit is too good. The same
is true, though to a smaller extent, of the moving five-year average. On
the other hand the simple nine-year average seems to have the sort of
properties we require to describe the general trend. We might have
guessed this at the outset by noting that the major fluctuations seem to
cover a period of about six years on the average so that a moving average
of at least six successive terms is required to smooth them out. See also
Example 26,5.

Approximate formula

26.20 By far the simplest kind of moving average to apply is the one in
which all weights are equal, and it is possible to simulate the accurate
formulae of (26.10) and (26.11) by repeated simple moving averages. For

TIME -SERIES 623

instance if we apply a simple average of threes to a series we have a series
typified by ; and if we apply a simple average of threes

to this series we have as a typical term

= ^[1,2,3,...] . . .(26.12)

The coefficients here follow more the pattern of (26.9) in that instead of
being equal they rise to a maximum at the middle member. We state
without proof that for many purposes great accuracy in the weights of
a moving average is not necessary, so that formulse of the kind of (26.12)
may be used as substitutes for the accurate formulae without serious loss
of efficiency.

Two formulae of general use in actuarial work are known as Spencer’s
15-point and 21 -point formulae. The weights are as follows —

Spencer s \S-point formula

Writing for a simple moving average of k terms, we have for this

formula

^[4] ^[5] [-3. 3, 4, ... ]

= ^[-3, -6, -5, 3, 21, 46, 67, 74, ... ] . (26.13)

Spencer's 2\-point formula

34 [5] ^[7] [-1.0, 1,2 ]

= 355(-1. -3, -5, -5, -2, 6, 18, 33, 47, 57, 60) . . (26.14)

These are accurate as far as third differences, i.e. they reproduce a cubic
exactly and wall provide a good approximation for higher order curves.
The advantage in using them lies in the fact that most of the arithmetic
can be carried out by simple summation. For instance, with (26.13) we
first of all find a moving sum of fours, then a second moving sum of fours
of the result, then a moving sum of fives of that result, and finally apply
the moving average of fives (—3, 3, 4, . . .) and divide by 320. This is
much more rapid than carrying out the moving average in one stage by the
weights given on the right hand side of (26.13).

624

THEORY OF STATISTICS

The statistician will rarely require closer fits than are given by these
formulae and frequently even they are too good m the sense noted in
26.19 A simple moving average often gives him what he requires if his
senes fluctuates , if it constantly moves in the same direction so as to
remain always concave or convex to the /-axis a simple moving average
will systematically under- or over-shoot the mark. Compare Exercise
26 14

26.21 We have chosen the number of points to which a polynomial is
fitted to be odd. This is convenient because in the contrary case either
the middle of the fitted range falls between two time-points or we have
to fit a polynomial asymmetrically Where, however, it is essential to
fit to an even number of consecutive points we can easily do so by a slight
modification of the technique Consider the case of data given by quarters
over a series of years To eliminate seasonal effects the natural thing
to do is to take a moving average of fours, but this gives us a set of values
which do not correspond to the time-points of the original data If,
for instance, the information is an average over each quarter, the
quarterly figures relate on the average to the middle of quarters and a four-
point moving average will give values at the end of quarters This may
be adequate for our purposes If not, we can “ centralise the trend
values by taking a four-point moving average and then a simple mean
(a two-point average) of the result. For instance, with a series starting
with the first quarter of 1948, a four-point average will give figures relating
to the end of June, the end of September, the end of December, 1948 and
so on. A simple average of pairs of the result will give figures relating to
the middle of August (the third quarter), the middle of November (the
fourth quarter) and so on. In effect, what this process amounts to is the

replacement of the scheme ^[1, 1, 1, 1] by Jl, 2, 2, . . ] as the reader can

4 o

readily verify. An example is given below (Example 26.2).

Elimination of seasonal effects

26.22 A great many time series, particularly in economics and
meteorology, are affected by the seasons Similarly, other natural
rhythms of shorter duration generate periodic effects such as the daily
rise and fall in temperature at a given spot or the variation in tides at a
port. Man-made periodicities may also appear, as m the change in the
nature of road traffic at week-ends, or the rise in current bank balances
at the end of the month. For smplicity we may term all such variations

seasonal*' where they correspond to indentifiable and strictly periodic
rhythms in the causative system even though the period is not one year.
The student should beware of regarding an oscillatory movement as

seasonal " (i.e. strictly periodic) merely because it presents some
appearance of regularity.

TIME - SERIES

625

26.23 Our object m considering seasonal effects may be either to get rid
of them in order to concentrate on the remaining variation or to isolate
them for separate study. Elimination is a simple matter if we are prepared
to extend our time-interval to cover a complete period of the seasons.
For instance, we can eliminate any seasonal effect in records of sheep
population by observing that population at a fixed date each year. The
same stage of breeding and slaughtering may not quite be attained on the
given date m different years but variations from it will be small and erratic.
Again, we may eliminate seasonal movements in rainfall by recording
only the total occurring in each year, the resulting senes of annual figures
containing no seasonal effects. Methods like these, of course, “ eliminate
seasonal movement only in the sense of choosing a longer time-interval
which covers one or more complete seasonal cycles ; they do not record
for each part of the year what the value of the series would be if the
seasonal part of the movement were abstracted, and to that extent they
sacrifice information.

26.24 To fix the ideas, consider a series of monthly prices of a commodity
such as eggs. This senes has a definite seasonal movement but also may
move from year to year independently of the purely seasonal effect. A
simple 12-point moving average is often sufficient to smooth out seasonal
variation but where enough data are available we may also take the
calculations further as in following example.

Example 26.2

The average monthly prices per 120 eggs in England and Wales in 1927
and January 1928 were as follows —

(1927)

Jan

Feb. Mar,

Apr.

May

June

July

Aug

Sept

Oct.

Nov.

Dec.

Price

(pence)

236

232 147

132

131

145

164

200

232

294

327

296

(1928)

jan.

286

The average of the prices for the 12 months of 1927 was 211 pence. The
monthly prices relate approximately to the middle of the month, (being
averages covering the whole month) and this average over the year there-
fore gives a range centred at the end of June. The average for the months
Feb. 1927-Jan. 1928 inclusive was 215 pence and this relates to a period
centred at the end of July. We therefore take as the appropriate value
for the middle of July the mean of 211 and 215, namely 213 pence. This
is the 12-month centred '' moving average or “ trend- value for July
1927.

The actual price for July was 164 pence and hence this price, as a
percentage of the '' trend- value is 16400/213=77.0. Calculations on
these lines for the years 1927-1936 are shown in Table 26.6.

TABLE 26.6. — Percentage relation of actual egg prices to 12-nionth moving averages and seasonal indices derived therefrom

Data from C T Houghton, J R Stai, Soc., 101 275.

626

THEORY OF STATISTICS

TIME -SERIES

627

In column (11) of this table is shown the average of the monthly indices
for each month ; and column (12) scales these figures down very slightly
so as to make them add up to 100*0. The results may be regarded as
an index of the purely seasonal part of the egg prices. The January
figure, for instance, indicates that on the average over nine years the
January price was 109-3 per cent of the trend- value for January, or that
seasonally prices are increased by 9-3 per cent in that month.

Let us now return to the prices for January-December 1927 quoted at
the beginning of the example. These include an element due to the seasonal
effect. Suppose we wish to eliminate seasonality in order to study whether
there was any real change in the price of eggs over the year. We
then divide the January pnce by 1 *093, the February price by 0-973 and
so on to obtain —

Corrected Jan. Feb. Mar. Apr. May. June July Aug Sep. Oct. Nov. Dec.

price

(pence) 216 238 209 213 203 202 194 195 211 216 207 212

These may be regarded as the prices corrected ” for seasonality. The

movement over the course of the year, apart from seasonal effects, is

obviously slight.

Change in price-level

26.25 As we have noted in connection with index-numbers special points
arise when our series are expressed in terms of money owing to the change
in the value of the unit over a long period. We may, therefore, wish to
remove from a series of prices a trend in the general price-level. This is
not the same thing as removing a trend in an ordinary series ; there we
are concerned with long-term changes in the numbers of units, whereas
here we are concerned with changes in the unit itself. The procedure
customary in such cases is to divide the actual price by an index of general
prices, or the price of gold, or some similar figure expressing the value of
money ; alternatively we may revalue on the basis of prices in some
standard year when our series relates to a basket of goods We have
noticed this latter process in Chapter 25. The former is illustrated in
Table 26,7. Column (2) shows the net national income per head of pop-
ulation in the United Kingdom. Column (3) gives an index of prices on
the basis of 1900=100. These figures are used to '' correct for pnce
changes " or to eliminate trends in prices to give column (4) which thus
provides figures for income per head of a more comparable kind.

The effect of trend elimination on other elements

26.26 The success or failure of a method of determining trend is to be
judged by results so far as the trend itself is concerned ; that is to say,
by whether it gives a sufficiently broad general picture of the movement
of the series for our purposes. But if our object is to eliminate trend in
order to study short-term movements in the series we have to be most

628

THEORY OF STATISTICS

TABLE 26.7.— Net National Income of the United Kingdom for certain years

Data from A R Prcbt, Economic Journal, 1948, 58, 31

(1)

Year

( 2 )

Income per
head at
current prices
£

( 3 )

Price index
1900=100

( 4 )

Income per head
at 1900 prices
col. (2)xl00/
col (3)

1900

42*7

100-0

42-7

1930

86-2

172-5

50-0

79‘5

161-5

49-2

77-1

157-1

49-0

80-2

153-9

52-1

83-1

155-0

53-6

87-6

157-1

55-8

93 2

161-5

57-7

97*6

169-2

57-7

98-3

171-4

57 4

From simple mne-poiat average (continuous line)
From a seven-point quadratic (broken line)

TIME - SERIES

629

careful that the residuals do not reflect the nature of the trend fitting
rather than any intrinsic property of their own. In no branch of
statistics do we have to guard so much against projecting our pre-conceived
ideas into the data by the technique of analysis we adopt.

Example 26.3

Figure 26.6 shows the residuals given by two of the three methods of
curve fitting derived in Example 26.1, the 9-point simple average and
the 7-point quadratic. (By residuals we mean the deviations of the actual
series from the trend values). Evidently the magnitudes of the deviations
ire very different in the two cases so that if we are interested in the size
of the residual fluctuation our result depends very much on which method
of trend-elimination we use. On the other hand, there seems to be a
regularity in the oscillatory movement which is common to both senes
so that any judgment as to the period of the short-term movement would
probably be very much the same whichever method of eliminating trend
we had adopted.

26.27 Suppose that a series consists of the sum of three components, a
trend, an oscillatory movement and a random element. Our method of
trend elimination by moving averages evidently acts separately on these
three components ; if, therefore, it eliminates the trend perfectly we
shall be left with residuals which are the same as if we had applied the
method to a series consisting of the sum of an oscillatory and a random
component. Let us consider the effect of the method on such components.

26.28 Consider an oscillation which is given by the terms of a sine-senes

Ut = sin^a-f

where a and A are constants. Such a series gives a harmonic wave of
period A. In most text-books of trigonometry it is proved that

^ sin{a+7r(fe+l)/A} . . .(26.16)

sin 7T I A

f,) .... (2AI5)

Thus a simple moving average of k terms will result in a sine-series with
the same period as the primary series but with amplitude reduced by
the factor

1 sin TTkjX
k sin tt/A

. (26.17)

If the process is repeated q times the amplitude is reduced by the ^th
power of this quantity.

If then k is large or itk /A is an integral multiple of tt, the expression

630

THEORY OF STATISTICS

(26.17) is zero or small. Thus the trend '' determined in the oscillation
is small and the residual only slightly affected. But if A is large and k /A
is small the term (26.17) is nearly unity (since sin d^d approximately
for small 0) and hence the residual will be very small, most of the primary
variation being eliminated as trend.

26.29 This is what we might expect on general grounds. If A /A is small
and A is large the oscillation has a large period, i.e. is a very slow one and
is treated as trend by the moving average. If the period is short compared
with k the residuals are only slightly affected.

In general, we may expect from this analysis that a moving average
will emphasise the shorter oscillations at the expense of the longer ones.
It is interesting to note that in some circumstances (26.17) may be negative
so that the oscillation in the residual may be even larger than in the
primary senes.

26.30 Consider, again, the effect of a moving average on a random
series with zero mean. To fix the ideas, consider a moving average
of fives. Two consecutive values of the trend would be typified by

^ (61+62+63+^4+^6) i (62+63+64+65+63) . ( 26 . 18 )

The variance of this series is \ var e and the covariance (since the residuals
are independent) is

~ E (62^+63^+642+652) = ^ var €

Thus the correlation between neighbouring terms is 4/5. Similarly the
correlation between terms 1, 2, 3, 4 members apart is 3/5, 2/5, 1/5, 0.
Hence the values of the '' trend will tend to be smooth ; and when we
subtract the trend from the original series we shall get a smooth component
on which is superposed a random series. The effect of trend elimination
is therefore to insert in the residuals a smooth component which, in
general, will exhibit oscillations. We have to take care, accordingly,
that when we detect '' oscillations in a series from which trend has
been eliminated by moving averages, the oscillations are not spurious.

Example 26.4

Figure 26.7 shows the results of a smoothing /g [5] [3] on a set of 35
random numbers which could vary from 0 to 19 inclusive. They were
obtained from the numbers on page 376 by reading two figure numbers
downwards and omitting multiples of 20, e.g. the first numbers are 9, 1,
3, 5, 7. The resemblance to the vague fluctuation of a trade cycle is
evident.

TIME - SERIES

631

Number of terms

Fig, 26.7. — Smoothing by a ^ [5] [3] average of a random series

Variate-differencing

26.31 As in the case of curve fitting (Chapter 15) the reader may wonder
how he is to find out in any particular case what sort of moving average
to use. If he is interested in trend the answer is as indicated in Example
26.1 . But if he is interested in residuals the answer is much more difficult.
We will indicate in broad outline a method which has as its object the
detection of random vanation e and the estimation of its variance and
which indicates at any rate an upper limit to the degree of the trend line.

Suppose a series consists of a polynomial of degree r plus a random
element. Then if we take first, second, third differences etc., the resulting
series consists of a polynomial of degree r—l, y— 2, etc., plus a residual
which increases in variance. We have, for instance, after the manner
of 24.15

E (Afif) 6/) =0 .

. (26.19)

var (Ae^) == E 6^)^

== 2 var e

. (26.20)

Similarly

•

var (A^e^) = 6 var e

. (26.21)

and generally

var (A^’e^) = ^ •

. (28.22)

632

THEORY OF STATISTICS

The effect of differencing is then to enhance the short-term movements
at the expense of the long term movements and in particular to multiply
the purely random element until it swamps all the others. (There is an
exception to this rule if the systematic part of the series has a short
period of two or less, for this is not reduced by differencing, as may be seen
by considering the series 1, --1, 1, -~1, etc.) This gives us a method of
estimating the variance of a random element superposed on a series
which can be represented (perhaps only locally) by a polynomial. The
variances of the first, second, . . . rth difference (or better, the second

moments about zero origin) are divided respectively by 2, 6, . . .

and if this quotient seems to be approaching a limit, the limiting value
provides an estimate of var e. Further, the degree to which we have had
to go is some indication of the degree of the systematic part of the curve.

Example 2^ S , — Consider again the sheep data of Table 26.1. A
calculation of the differences would proceed as follows —

2203

2360

2254

2165

2024

-157

106

141

-263

-52

-280

etc.

The sums of squares of the differences A'' are shown in the following
table. Column (3) shows the number N of terms on which they are

based, and column (4) the ratio E (A'')2/iV^^^^j, that is to say the ratio

which we expect to tend to the variance of e.

TABLE 26.8. — Variate-difference analysis of the data of Table 26.1

(l)

Order of difference
r

(2)

Sum of squares of

(3)

Number of terms
in sum N

(4)

Column (2)

K")

499,356

3468

614,333

1442

1 1,195,999

854

3,037,326

629

8,883,670

518

27,735,006 j

448

90,957,010

402

310,670,360

371

1,110,091,780

357

4,043,696,988

347

TIME - SERIES

633

We also find, for the original series

'Liut) - 135,537, S(w,2) = 267,800,918.

whence we have for its variance

= 272,229.

A comparison of this figure with the fourth column of Table 26.8 shows
that the variation is very substantially reduced by the first two or three
diferencings. We should be justified in concluding that the data can be
represented locally by a polynomial of the third or fourth order, e.g. by
a moving cubic or quartic and that the error e (regarded as superposed
on this systematic representation) has a variance of about 500.

What we have said above about the adequacy of a trend line is in
no way affected by this result. The present example tells us that if the
data consist of a polynomial plus a random element, there is no need
to seek for a polynomial of degree higher than four. It indicates that
we should be wasting our time in trying to fit quintic or higher order
curves (or in using moving averages based on quintics, etc.). It does
not say that a quartic is the best trend line for the purposes of a broad
description of the trend , a simple curve might be more suitable in
particular circumstances.

SUMMARY

1. For descriptive purposes the most general form of univariate time
series may be regarded as composed of trend, short-term systematic
movement and random or haphazard components.

2. This analysis sometimes corresponds to different causative systems,
but not always so.

3. A convenient method of trend determination is to use moving
averages. The weights can be determined by least squares and approxi-
mations to the exact weights are legitimate and useful.

4. Seasonal effects, i.e. movements occurring in a strictly periodic
manner, can be removed or isolated by a special method.

5. Moving averages may distort short-term components and generate
spurious oscillatory movements in random components of a time-series.

6. Variate-differencing can be used to estimate the variance of the
random component of a ser,ies on the assumption that the other components
can be represented (at least locally) by a polynomial in time and that
no periodic movement is present with a period of two intervals or less.

EXERCISES

26.1 Determine a trend line by a simple moving average of nines in
the data of Table 26.1 for the years 1905 to 1939.

634

THEORY OF STATISTICS

26.2 The values of a series are plotted on a diagram in the usual

way with t as abscissa. The points corresponding to ^^^2 ^.re joined

and the line joining them bisected, giving an ordinate of say, v^. The
process is repeated by bisecting the line joining and W 3 to give ;
and so on along the series.

The procedure is repeated with the series • • • ^n-i to give a senes

w^, , , Wn-^. Show that (w^ 4 - 2 «^< 4 -i+^«+ 2 )- Examine the suitability
of this procedure as a method of determining a trend line in the data of
Table 26.2.

26.3 The following are the figures for the infantile mortality rate in
England and Wales (deaths of infants under one year of age per 1,000
live births) —

Year

Rate

Year

Rate

1922

1935

1940

1930

Fit a simple moving average of fives to this series and apply a further
simple moving average of fives to the result.

26.4 The following is the rainfall in inches in England and Wales for
certain months —

Avearge
1881 -1915

1943

1944

1945

1946

Jan.

2-99

6*2

3*3

3*4

Feb.

2*57

1*8

1*7

3*1

3 4

Mar.

2 67

0-9

0*5

1*3

1*5

Apr.

2 12

1*4

2*2

1*7

1-8

May

2-30

3*2

1*5

3*2

3-0

June . . ;

2-44

2*3

2*5

3-2

1 3-4

July .

2-87

2*2

2*8

2-6

3-1

Aug

3*35

3*2

2*8

i 5-4

Sept. . . 1

2*54

3*4

4*2 1

2*5

4-9

Oct.

3'97

3*4

4*5 '

4 1

! 1-5

Nov.

3*49

2*7

6*1 !

0*8

6*2

Dec.

3*92

2*1

2*8 '

4*1

4*0

Annual Total

35*23

32*8

35*3 i

32*8

41-6

TIME - SERIES

635

Using the average of the period 1881-1915 as a norm derive monthly
index numbers, for the period 1943-6 of the rainfall “corrected" for
seasonality. Graph your results.

26.5 If the smoothing formula 2 \ [—2, 3, 6, 7, . , . ] is applied to a
random series, find the correlations between members of the smoothed
series 0, 1, 2, 3, 4, 5, 6 members apart.

26.6 Construct ten terms of the series whose value at time t is
for jf— 0, 1, ... 9. Verify that the formula

k [- 3 , 12 . 17. ... ]
gives an exact fit to such a series.

26.7 Take the random digits of 16.30 as random numbers which can
vary from 0 to 9 with equal frequency in the long run. Take a simple
moving average of threes of the first 50 terms, then a simple moving
average of five of the resultant, then another simple moving average of
five of that resultant. Note the appearance of smooth series fiom the
repeated averaging.

Write down the coefficients of the smoothing process if carried out in a
single stage.

26.8 The following is an index number of the price of lead from 1926
to 1945 together with the “ Statist " wholesale price index for the period.
Construct an index of lead prices " corrected " for changes in the whole-
sale price level.

Year

Index numbers

Year

Index numbers

Wholesale

prices

Lead

Wholesale

prices

Lead

1926

125

157

1936

122

125

102

121

119

109

114

117

1930

1940

128

127

142

129

2 ;

151

129

3 ^

155

129

160.

129

164

142

636

THEORY OF STATISTICS

26.9 For a series in which the values are represented by a cube or lower
power of the time variate t show that, if is written in brief for a
simple moving average of k terms,

^ m - 'V [^]

gives an accurate trend line. Hence show how, by two simple moving
averages, we may obtain a trend formula which will be correct to the
third degree in the fitted polynomial.

Obtain the formula when A =5, ^=3 in the form

lo [- 1 , 4 , 4 ]

26.10. By considering the senes (^—2)^ . . . (^+2)^ show that

the formula

[ 6 , - 46 , 1 + 66 , - 46 , 6 ]

accurately reproduces a cubic curve for any value of 6. Show further
that if this formula is applied to a random senes the correlation between
neighbouring members in the resultant '' trend ” is

- 86(1 + 76 ) /( 7062 + 126 + 1 ).

26.11 The following are the quarterly index numbers of wholesale prices
in the U.K. published by the '' Statist

Year

Quarter

2 3

1928

122

125

118

117

119

114

109

1930

105

By a “ centred '' moving average of four calculate a quarterly index
corrected for seasonal effects.

TIME - SERIES

637

26.12 If S is the “ central ” difference defined by

6ut = Ui+i—Ut-l

show that to third differences,

where ^ [H] stands for a simple moving average of h.

26.13 Verify equations (26.10), and show generally that the same
formulae are reached for polynomials of order 2p-\-l as for order 2p.

26.14 The value at time t is given by V(^ /lO). Sketch the series

from ;{=0 to ^=100 and show that the '' trend '' determined by a simple
moving average is always less thati the actual value of the series.

CHAPTER TWENTY-SEVEN

TIME-SERIES-(2)

27.1 In this chapter we shall consider the short-term and random
components in time-series, and shall suppose either that our series have
no trend present (as in Tables 26.2 and 26.3) or that, if trend was originally
present, it has been removed. Our series will then fluctuate more or less
irregularly about some central value which we may regard as the mean
of the whole series ; and our problems are to detect and to investigate
the nature of the components of such fluctuation.

Tests for randomness

27.2 Let us first consider what kind of series we are likely to obtain if
the variation is entirely random, i.e. if successive values are independent
and the series may be considered as the chance arrangement of a sample
from some unknown population. Two features suggest themselves as
natural measures of departure from this situation, {a) the occurrence of
peaks and troughs in the series and (b) the correlations between neigh-
bouring members.

27.3 A member of a series Uf is said to be a peak ” if <u^>

and it is a trough if Ut_,^ >Ui< In either case it is a '' turning-

point and the interval between turning points is called a phase

If two or more successive values are the same and are greater than neigh-
bouring values we regard them as determining one peak situated in the
centre of the range of equal values ; and so for troughs.

It may be shown that in a random series of n terms the mean and
variance of the number of turning points p are given by

/h'm = §(«- 2 )

(27.1)

/liip) =

16»-29

(27,2)

These results are independent of the distribution of the parent population
of values of the series and therefore have a considerable generality. As
n becomes large the distribution of p tends to normality fairly quickly.
For large n the average number of turning points per unit interval is
2/3 and the average phase (the average distance between such points)
is therefore 1 *5. Hence the average distance between peaks (or between
troughs) is 3, and this is what we expect to find in a random series.

638

TIME -SERIES — (2)

639

Example 27.1

Consider the data of Table 26.2. If n=19 we have, from (27.1) and
(27.2), a mean value of 11.3 and a variance 3*05 for p. The actual number
of turning points in the table is 9. The deviation from the mean, 2*3, is
less than twice the standard deviation of about 1*75 and we conclude that
this evidence is not significant of departures of the series from randomness.

On the other hand in Table 26.4 where n=48 the mean and variance of
p are 30*67 and 8*21. The observed value of p is 14 which differs from
the mean by more than six times the standard deviation. We cannot
therefore regard the series as random.

Serial correlation

27.4 The coefficient of product-moment correlation between the neigh-
bouring members of a series is called the autocorrelation of order 1 ; and
similarly the correlation between members (^ — 1) apart is called the auto-
correlation of order k. Thus

, . . (27.3)

V{var Ui var

These functions are very important in the theory of oscillatory time-series
and have applications far beyond the purpose for which we are now going
to use them. Where it is important to distinguish between the values
derived from a parent series and those from a sample we shall call the
latter serial correlations and denote them by The contrast between
auto and serial (of Greek and Latin origin), as between p and r, accords
with our usual practice of denoting parent values by Greek and sample
values by Latin symbols.

This usage is not universal. Some writers use “ autocorrelation ” to
denote the correlation of members of a series among themselves, whether
in population or in sample, and '' serial correlation to denote the correla-
tions between different series.

27.5 In a long series var u^ and var
(27.3) becomes

„ _ COV [Ut,

var Uf

are practically identical and

. . . . (27.4)

For short observed series it is better to take the variance of the whole
series (calculated from n terms) as the estimate of var u although the
covariance is based on only w— ^ terms. Similarly it is better to calculate
the deviations of u from the mean of the whole series in determining the
product-sum of and Then, if the members of the series are

measured about the mean of the whole set of terms we then have

* n—k

n—k

S {U,

I«1

(27.5)

640

THEORY OF STATISTICS

27.6 Now if a senes is random the theoretical value of p* is zero for any k
other than ^=0. We may therefore use the departure of the serial
correlations from zero to test departure of the series from randomness.
We state without proof that for large n the variance of in a random
series is approximately

( 27 . 6 )

Example 27.2

Table 27.1 shows the values of the residuals of the sheep series of
Table 26.1 when trend has been eliminated by a simple moving average
of nines.

The value of for this series of 65 terms is 0*595. The standard error
in a random senes, from (27.6) is 1/^64=0*125. The observed value
is therefore significant and w^e conclude that the residual series cannot
be regarded as a random one.

TABLE 27.1. — Residual values of the sheep series of Table 26.1 after elimination of
trend by a simple nine-point moving average

Year

Residual

(10,000)

Year

Residual

(10,000)

Year

Residual

(10,000)

1893

+ 34

1915

+ 19

-103

+ 128

-104

+ 97

+ 141

- 15

+ 69

+ 60

- 23

- 29

- 20

+ 17

-174

+ 12

+ 71

-107

+ 82

1900

+ 3i>

-142

+ 130

+ 16

-109

- 14

- 27

- 23

-166

- 32

+ 60

-179

- 49

+ 121

- 84

- 61

+ 94

+ 38

- 52

- 25

+ 97

- 24

- 90

+ 8

+ 68

- 75

- 5

+ 141

+ 72

-105

+ 119

+ 152

- 99

+ 66

+ 112

+ 35

- 52

- 64

+ 159

-117

- 87

92 1

+ 167

- 61

The calculation of serial correlation is rather a tedious process but
help may be obtained by the following device. The series of n terms
is written down vertically on each of two slips of paper, the spacing being
equal on the two slips. This can very conveniently be done on a tabulator
^th a split keyboard. To calculate the first product-sum we pin the

TIME - SERIES — (2)

641

slips so that the first term on the right-hand slip is opposite the second on
the left-hand slip and so on all the way down. For most series the
difference of two terms which are opposite can be obtained mentally by
subtraction, squared and set up on an adding machine. The sum of
squares of differences is thus determined and the cross product
derived from the simple identity of the type

2^{xy) == 2

with the aid of which is obtained without difficulty.

27.7 Tests of the randomness of a time-series are often unnecessary
because it is obvious from inspection that the series is systematic to some
extent. The two tests we have given, however, may be applied when
there is any doubt and will usually be sufficient to settle it. Suppose
now that we have decided that our series is not random. Some part
at least of the oscillatory movement then requires explanation. To set
up models which will reproduce the behaviour of oscillatory series is one
of the most difficult outstanding problems of current statistical theory
and it would be quite beyond the scope of this book to give an account
of even what is now known, incomplete though that is. What we shall
do is to describe and illustrate two techniques, one classical and one new,
which offer the most promise.

Pcriodogram analysis

27.8 The reader who has an acquaintance with elementary physics is
probably familiar with the way in which the motion of many oscillatory
physical phenomena (tides, violin strings, pendulums and so forth) can
be represented as the sum of a number of pure harhionic waves each
of which can be represented by a sine or cosine term. The motion of a

pure oscillator in time is expressible as a term A sin

where A is

the w^avelength and A the amplitude ; and oscillatory phenomena can
often be represented by a sum of such terms —

sin sin . - +etc. (27.7)

Light itself is a phenomenon' of this kind and Newton's classical experiment
with a prism in splitting white light into a spectrum may be regarded as
an analysis of a complicated periodic phenomenon into simple terms
each with its own colour ” or wavelength.

27.9 Aware that many physical phenomena can be described by series
of type (27.7), early investigators of economic and meteorological time-
series were led to inquire whether the same methods could be used to des-
cribe them. The basic idea was that the senes could be regarded as the
sum of a number of strictly periodic terms plus, perhaps, an error of

642

THEORY OF STATISTICS

observation. This search for strict periodicity has not been very successful.
The model on which it is based requires that, apart from casual errors, the
peaks and troughs shall recur at equal intervals whereas in economic
series at least crises certainly do not recur with strict regularity. Further-
more, the model presupposes that errors '' behave like errors of observa-
tion, that is to say, that they occur to disturb the observation at a particular
moment but do not alfect the subsequent motion of the system. Now
in economics and meteorology, at least, it is more plausible to suppose
that when something happens to disturb the system, the effect of that
disturbance is integrated into the future motion of the system and becomes
part of it. The model of superposed harmonics is not therefore a very
plausible one. Nevertheless there are branches of our subject where
analysis into harmonic components (i.e. sine or cosine terms) is useful
and this chapter would be incomplete without some reference to it.

27,10 The process of searching for the ‘periodicities in a time-series by
harmonic analysis can be compared to the tuning of a radio set. We
correlate a number of series with known wavelengths with the given
series and if they are “ out of step '' with the wavelength of the series
the result is a low intensity ; but when we come into tune with that
wavelength, there is a high intensity of correlation ; and hence by con-
sidering the various intensities we can discover whereabouts the true
wavelength lies.

To put it more accurately we select a trial wavelength /i and form the
sums

^ ? S cos . . . . (27.8)

« j=i /« '

jB = - S sin — ^ . . . . (27.9)

and write

52 =^ 2+52 .... (27.10)

Then S is known as the intensity. Apart from constants the numbers
A and B are the covariances of the series with the tnal sine and cosine
terms.

Now suppose that the series is in fact given by

Ui = a sm

.(27.11)

where is a term uncorrelated with the trial period. Then
A

2a ” . 2777 2ni

S sm cos ——
n A ji

S sin aj cos

TIME-SERIES — (2)

643

where a = 27r/A, = 2n jfi

= ^ S { sin (a-^); + sin }

^ flf sin {|(«— /g)w} sin U(«-^)(w+l)}
n{_ sin {i(a-/?)}

+ sm-U(a+A)> J ■

with a similar expression for B in which sin {n +1)} is

replaced by a cosine. Now for large n this is small unless the term in
square brackets is large, that is, unless a ~ or a -)- /? is small. In
this case, neglecting the term of order 1 /n we find

/12 4 _r 2 = ^ sin^ {|(a - fi)n}

^ ' sm^{Uoc-m

and' since, for small 6, sin 6 = d approximately, sin{| {a — fi)n} =
I (a ~ P)n and we have

+ (27.13)

Thus S remains small unless a is nearly equal to /? (and hence the trial
period [i is near to the real period A) in which case S is equal to the constant
a and gives the amplitude of the term.

27,11 To calculate the sums A and B, suppose in the first place that ft
is an integer. Write down the series in rows of [i thus :

% . . . W/i

W/i+i • • • ^2/4

^(p-“i)/4+i • * • (27.14)

Totals ... fUfi

We continue writing down the rows until there are fewer than [i terms
left, the extra terms being neglected. The number pii is then as near as
we can get to n in multiples of ji and may be denoted by N.

The sum

27r , 47r ,

w. cos h ^2 cos h •

m^tcos

2/^7r }

"7” J

(27.15)

is then the sum A of (27.8) for JV terms. Similarly we have a formula
for B with sines instead of cosines.

In practice, of course, we do not actually form such a table as (27.14).
The sums may be formed direct, from the series on an adding machine
by adding every p\ix member, starting in turn at Wj, and so on.

TIME - SERIES — (2)

645

27.12 The graph of 5 as ordinate against fi as abscissa gives us a feriodo-
gram and the whole process of analysis is known as periodogram analysis.

Example 27.3

Perhaps the most famous (and certainly the most exhaustive) example
of a periodogram analysis is the one carried out by Lord (then Sir William)
Beveridge on a series of index-numbers of wheat prices constructed by
him for a period of about 300 years. Figure 27.1 shows the resulting
periodogram.

Beveridge worked out the intensities for many trial periods which are
not integral. The method is the same in essence as that of 27.11. For
instance, if /i = 10/3 we write down the series in rows of 10 and multiply

the sums . . . m^^ by cos — , cos — , . . . etc., in forming A, There

were, in fact, many more trial values for lower values of fi than we have
been able to show on the diagram.

The interpretation of a periodogram like this is very difficult. Beveridge
himself was inclined to attribute significance to 18 or 19 major peaks, and
was only following the practice of the physical sciences in doing so. It has,
however, subsequently been shown that three-quarters of the peaks are
explainable as sampling effects. In fact, it may be shown that if v is
the variance of the series the chance that exceeds ivKjn in value is
and hence if q trial periods are picked out at random the chance that one
at least should exceed 4vk jn is

1— (1-5--^)?

On the basis of this criterion, the peak at /^==15*25 is significant
and possibly those at /i=5-l, 12*8, 17*3 and 20*0 are significant, but
no more. More recent researches on the periodogram for an autoregressive
series (27.13 below) indicate that it may be smoothed and on this basis
the peaks at 5 • 1 and 15*25 alone would be significant. But we shall
have to make these statements without proof and, indeed without adequate
discussion, merely to warn the student to mistrust most of what he finds
in the literature on the penodograms of time-series. Different writers
have been led to claim the existence of cycles of all kinds in economic
and meteorological data. A reconsideration of the data would probably
show that none of these cycles exists in the sense of being strictly periodic,
at least in economics.*'

Autoregressive series

27.13 A more modern approach to the subject attempts to take into
account the point we noted in 27.9, namely that when a disturbance
occurs it is integrated into the motion of the system. Instead of regarding

* For some further discussion and tables to facilitate the performance of a periodogram
analysis see Kendall, ContnhuUons to the Study of Oscillatory Ttme-Series, 1946, Cam-
bridge University Press.

646 THEORY OF STATISTICS

our system as oscillating like a pendulum (the only departure from
harmonic motion then being in the errors of observation) we shall consider
it as swinging like a pendulum subjected to a continual stream of shocks,
as for instance if it were pelted by small boys at random with peas. The
pendulum will continue to swing backwards and forwards, but not
regularly so. The times between its swings will not be constant nor will
it always swing out to the same extent. In fact it will behave very much
as many oscillatory time-senes are seen to behave, which is our main
justification for introducing this model for study.

27.14 We shall suppose that the motion of the system is determined
by two factors : (a) a group of internal properties such as elasticities and
constraints which determine how the system moves if left to itself and
(b) a series of external shocks. We shall further suppose that the existence
of factors in the first group can be expressed by saying that the value of
the series at time ^ is a linear expression in values at previous points of
time. We shall then have equations such as

^t+i .... (27.16)

where ju is a constant and e represents the external disturbance ; and

^<+2 “ • • • (27.17)

where again oc and jS are constants. Such series are said to be aulo-
regtessive because (27.16) and (27.17) may be regarded as regression
equations of one term of the series on previous terms. More elaborate
systems can, of course, be devised but these two simple cases are all we
shall consider.

Values of t

Fig. 27.2.— Graph of the series of Table 27.2

Values of Series,

TIME - SERIES — (2)

647

Fig. 27.3.— Graph of the values of Table 27.3

TABLE 27.2. — ^Values of series ut+i = 0 7 -f
Where is a random normal variable with zero mean
From Kendall, 1949, Btomdnka, 36 267

Number of term

Value of senes

Number of term

Value of senes

2*390

0*546

0*985

-0*886

-0*655

-1*321

-0*679

-1*014

-0*044

-2*254

-1*457

0*582

-0*731

0*272

-0*724

0*358

-1*567

0*981

-1*654

-0*497

-2*416

-1*078

-2.*821

-0*318

-0*701

-0*597

-1*515

1*697

-2*112

2*585

16 i

-1*602

0*170

-1*805

0*497

-1*624

0*437

-1*060

1*554

-0*022

1*474

648

THEORY OF STATISTICS

Example 21 A

To show how series of this kind behave, we give in Table 27.2 and Figure
27.2 the graph of a series of type (27.16) with/4==0'7, the values of ebeing
random numbers chosen from a normal population.

In Table 27.3 and Figure 27.3 we show similarly the graph of a series
of type (27. 1 7) with a = — 0*5 where e is a random variable chosen

by selecting random numbers from range —-9 * 5 to +9-5.

The irregular occurrence of peaks and troughs in such data is quite
clear from the diagrams.

TABLE 27.3. — Values of series 1-1 — 0 5 w; +

Where 6^+2 is a rectangular random variable with range— 9 5 to 9 5, rounded off to

nearest unit

From Kendall, 1944, Btometnk'i, 33, 105

Number
of term

Value of
senes

Number
of term

Value of
senes

Number
of term

Value of
senes

_ 4

-13

- 5

- 6

- 9

- 4

- 5

- 1

- 8

- 6

- 1

- 3

- 4 !

- 2

- 4

- 7

- 1

- 2

- 3

- 1

- 5

- 8

-11

- 3

- 8

- 3

-10

- 5

-16

27,15 Consider now the series of (27.16) in the form

.... (27.18)

where e has zero mean (and hence so has u) and successive values of e
are independent. It will be clear from the series that Ut involves e^^i,
etc., but not Let us then multiply (27.18) by Ui^j^ and

TIME - SERIES — (2) 649

sum over all values of u. Since cov var u where

Pk+m is the (A+w)th autocorrelation, we have

{Pk+i—PPk) var u = cov(€i+j, Ut,j,)
and since the covariance on the right vanishes for ^> — 1 we have

Pk+l PPk ^ i . • . • (27.19)

In particular when ft = 0

Px= P • • • • (27,20)

and hence

. . . .(27.21)

We may note from (27.20) that only values of ji not greater than unity

are admissible. If /i were greater than one the series would increase in

amplitude and explode to infinity.

27.16 In a like manner, for the series of (27.17)

«,+2+a«t+i+/?M, = e,+2

we have, on multiplying by and summing over u

ft>— 2 . . . (27.22)

In particular, for ft = — 1, ft = 0 we have

Pi(l+/^)+a-0

p2+a/>i+A =0

leading to

a = - (27.23)

(27.24)

It may be shown by the theory of finite difference equations (we omit
the proof) that the solution of (27.22) is

_ sin (ftg+^)
sin ^

(27.25)

Values

650

tHEORY OF STATISTICS

TABLE 27.4.— Serial correlations of the sheep data of Table 27.1

Order of
correlation
k

0 595

-0-142

-0-381

-0-151

-0 172

-0-118

-0 601

-0-186

0-173

-0 537

-0-128

i 0 343

5 !

-0-138

0 052

0-352

0-144

0 276

0-154

0 203

0-439

-0-203

0-118

0-293

-0 456

0 006

-0-074

-0-415

-0 078

-0 359

-0-184

1*0

Fig, 27.4.— -Coirrelogram of the sheep population data of Table 27.1

TIME - SERIES — (2)

651

where

^ = VA positive sign
cos 6 = -~a/(2'v/^)

tan ^ ^

( 27 . 26 )

Here again there are restrictions on the constants a and The latter
must be positive for p to be real and since cos 6 is not greater than unity
a2<4/?. Further, since cannot exceed unity p cannot do so. Hence
p must be positive and not greater than unity and a must be not greater
than 2 in absolute value. If these conditions are not obeyed the series
will not oscillate within bounds but will diverge to unlimited values

27.17 The results of 27.15 and 27.16 serve two main purposes. If we
know that the series are of the linear autoregressive type, (27.20), (27.23)
and (27.24) — and similar equations for more complicated series — enable
us to estimate the constants fi, a and yff in terms of the autocorrelations
which, for large samples at least, we may take to be the observed serial
correlations. Secondly, the laws obeyed by successive autocorrelations
as exemplified in (27.21) and (27.25) enable us to judge whether given series
are of the autoregressive type.

The correlogram

27.18 The graph of the autocorrelation as ordinate against k for
abscissa is called a correlogram. Since p-ic=Pk we draw it only for non-
negative values of k. Table 27.4 and Figure 27.4 give the serial correla-
tions and the correlogram of the sheep data of Table 27.1. There is a
marked oscillatory movement which may be compared with Figure 27.5,
giving the correlogram of the artificial series of Table 27.3.

27.19 Equation (27.21) shows that the theoretical correlogram of a
series of the autoregressive type (27.18) will be a simple curve decaying
from unity at ^=0 to zero at oo, the ordinate at each point k being

times the ordinate at the previous point. On the other hand equation
(27,25) shows that the theoretical correlogram of the series (27.17) will
not otily decay according to the factor p but will also oscillate. This
so-called damped harmonic is illustrated in Figure 27.6.

These theoretical forms, however, are reproduced only approximately
by series of finite length, as Figure 27.5 illustrates. The correlogram
oscillates and its earlier terms damp out, but there comes a point when
no further damping appears. This failure to damp must be regarded as
a sampling effect.

652

THEORY OF STATISTICS

TABLE 27.5.— Serial carrelations of the artificial series of Table 27.3

Order of
correlation
k

0-70

^0-05

0-05

0-29

•~-0*17

-0 12

0-01

-0*27

-0-28

-0*17

~0-31

-0*43

-0*27

-0*30

-0 57

-0-25

~0-18

-0-56

-0-13

0*12

-0-26

0-07

0-29

0-02

012

0-33

0-17

0-05

0-22

0*27

Fig. 27.5. — Correlogram of the artificial series of Table 27.3

TIME -SERIES — (2)

653

27.20 Let us return to the scheme of harmonics represented by (27.7).
It may be shown that for the series

“ ' 27r.\ ,

H-e,

Mj = h A/ sin

the correlogram is given by

cos

~ I,(Af)+2 var e

. (27.27)

provided that e is independent of the harmonic terms.

Thus to any term with amplitude the original series there corre-
sponds a wave of amplitude Af /{Af +2 var e) in the correlogram which
is undamped.

Theoretically, then, the correlogram should give us a method of dis-
criminating between the scheme of superposed harmonics and the auto-
regressive scheme. In one case the oscillations in the correlogram do not
damp out, in the other case they do. In practice, for short series, the
discriminating power of the correlogram is not very high, owing to the
failure of autoregressive correlograms to damp out for sampling reasons.
Nevertheless an examination of the correlogram is often a very good way
to start an investigation into the generating model of a given system.

Example 27,5

Consider again the data of Table 27.4. Taking the observed serial
correlations as the parent values we have

= 0*^595, rg = — 0*151.

Hence, from (27,23) a (the estimate of a) = —1-060
and from (27.24) h { „ „ /?) = +0-782

654

THEORY OF STATISTICS

If the series can be represented by the three term linear autoregressive
scheme then that scheme is

^<+ 2 — 1 * * 782 ^/^ = 6 ^ 4-2

It is natural to wonder whether a three-term scheme is adequate and
whether more terms may not be required. The question may be answered
by the calculation of partial correlations. The following are the partials
of the present series in our usual notation, 13.2, for instance, denoting

the correlation between and Uf+i

when

is constant.

Order of partial

Value of

correlation

partial

12.

0-595

0-6460

13.2

-0-782

0-2509

14.23

0-097

0-2485

15.234

-0-183

0-2402

16.2345

0-031

0-2400

The product 1 in the last column measures (12,20) the closeness
of the representation of the series and it is clear that little extra accuracy
is gained by taking more than three terms, which will account for 75
per cent of the variation.

27.21 It may be added that for the purposes of detecting oscillatory
movements by correiogram analysis shortness '' is a relative term. Even
series of 400 terms are sometimes short in the sense that the correiogram
after the tenth serial correlation or so does not damp out after the manner
of Figure 27.6. A consideration of the magnitude of the variance of serial
correlations in a random ‘series, ll{n—k), will show why this is so; for
n—k of the order of 100 the standard deviation is 0 - 1 and values of r as
great as 0-3 are not impossible. What does appear to be true in practice
is that even if the amplitude of the oscillations does not decay quickly,
the swings in the correiogram conform to the period of the generating
scheme as in Figure 27.4.

27.22 We conclude the chapter with a brief account of some of the
properties of the autoregressive schemes of (27.16) and (27.17), Let us
note as a preliminary point that such schemes will always give an
approximate representation of the series in the sense that a regression
line will always approximately represent the data to which it is fitted.

From relations such as

lit + U

~ 6/+// +/^"( +//

. (27 28)

TIME - SERIES — (2)

655

we see that the series may be regarded as a moving average of infinite
extent of the series of e’s.* The weights decrease and the contribution
to lit of is proportional to /t*. that is, the contribution of the past
is less and less important as it becomes more distant, which is what we
should expect. We have directly from (27.28), when the e’s are in-
dependent,

var Ut = (1 . . . ) var e

= .... (27.29)

expressing the variance of the series in terms of that of the disturbance
function e. If fi is near unity the variance of u may be much larger than
that of 6.

27.23 In a similar manner it may be shown that for the three-term
series (27.17) the solution of Ut, apart from terms which will have damped
out of existence if the series was begun a long time ago, is also a moving
average of the e's and is given by

= E .... (27.30)

6(7 .. . (27.31)

These weights are themselves oscillating and damped, like the correlogram.
It may also be shown that

var u 1

var e "" (1-/?){(1 -(-/?)

which reduces to (27.29) when a==/i, /?=0 as it should.

Example 27.6

In Example 27.5 we found for estimates of a and y? the values of —1 -060
and +0*782 respectively. Substitution in (27.32) gives

varw == 3*778 var e

Thus of the total variation of the series var e represents about 1/3-778
or 26 per cent, which agrees with the estimate given by 1 — in Example
27-5 within one per cent.

’•'The series of (27 28), to be a complete solution, should have added to it a term
AiA.i where A is an arbitrary constant. We suppose, however, that the series began a
long time ago so that this term has damped out of existence, /4 being less than unity.

656

THEORY OF STATISTICS

Example 27.7

The sunspot data of Table 26.4 are an extract from a larger series
beginning in 1749. An analysis of the series of 176 terms ending in 1924
Yule, Vhil, Trans,, A, 226, 267 gave the following —

a = — 1*342, h = -|-0'655

Partial correlations indicated that this series was adequately represented
(about 80 per cent) by a three-term autoregressive scheme and no im-
provement would be given by further terms. Thus it appears th at the
series can be regarded as autoregressive with a damping factor ^ = \/0* 655
=:0*81 approximately. The period in the correlogram {d of equation
(27.26) is given by

^ 1-342

^ - 2^0-655 ~

0-829

giving 33° approximately. Thus the period of the correlogram is
360/33=10*6 years. The series itself has no single ‘"period” because
the interval between successive peaks and troughs varies.

The period ” of an oscillation

27.24 From what we have said above it will be clear that for auto-
regressive schemes we cannot speak of the period of the series. There
wilbbe one period in the correlogram for the three-term case of (27.17) —
or more with more elaborate schemes — and perhaps we might call this
the autoregressive period. But it does not necessarily correspond to
the mean-distance between peaks in the series itself and in any case the
distances between peaks vary. The same is true of the distances between
“ upcrosses ” or “ downcrosses ”, namely points where the series (measured
from its mean) change sign from negative to positive or vice-versa.

The autoregressive period of (27.17) is given by 27r jd where as in (27.26)

cos 0 = -a/2i/A .... (27.33)
Now consider the series of values.

yi = Ui+i—Ui+2

We have, since the mean values of x and y are zero

. (27.34)

var Xi =.var Wf+i4-var Ut—2 cov (Wf+i,

= 2 var u (1
= yar^/*
and

cov {Xt,yt) =cov {Uu «t+ 2 )+var Ut+i

— cov We+j)— cov Ut)
===VBiuil^2pj^+P2)

TIME - SERIES — (2)

657

Hence for t, say, the correlation between and we have

1 — 2p^+P8
2{l-p,)

. (27.35)

Now suppose that x and y are normally distributed, as will be the case
if u is normal. The relative frequency with which x and y are positive
(i.e. Ut+■^^>u^ and so that is a peak) is then the relative

frequency in a bivariate normal distribution in the positive cell among
the four into which it is divided by s;=0 andy=0. This, by Sheppard’s
theorem (Exercise 10.4) is given by / where

T = cos (1—2 f)n
= —cos 277 /

so that

1 // = 277/cos-^(— r) .... (27.36)

and this gives us the mean distance between peaks.

27.25 For the autoregressive scheme (27.17) we have in virtue of (27.23)
^and (27.24)

r = i(l+a— /?)

and thus

mean distance (peaks)

277

co.s-iH(i-h^-A)}

. (27.37)

. (27 38)

which is not the same as (27.33).

Example 27.8

Consider a series for which a=— !•!, /?=0-5. From (27.38) we find
for the mean distance between peaks

r = J(l-M+0-5)=--()'3
cos-iO-3 = 72-54°, l// = 4-96.

In a series of 480 terms constructed according to this formula Kendall
(J. Boy. Statist. Soc., 1945, 108, 93) found an observed value of 5-05, in
excellent agreement.

On the other hand for the autoregressive period, from (27.33)
cos 0 = 1 • 1 /2V0-5 = 0-7778, 6 = 38.9°
giving for the autoregressive period 360/38-9 =9-3 units.

27,26 Two final comments ;

(«) We have emphasised that for certain types of oscillatory series
the idea of a single period or set of periods in the strict sense may be
inappropriate. The student who is» interested in oscillatory movements
should accustom himself to think of the distribution of distances between

658

THEORY OF STATISTICS

peaks or upcrosses as expressing its oscillatory behaviour, in the same
way that he thinks about a distribution of frequencies as characterising
a population ;

(b) For series of such a type the existence of the random variable e
means that there is a limit to the accuracy with which we can predict
the behaviour of the series. The autoregressive scheme will account,
at least approximately, for a certain amount of systematic movement
expressible in terms of the constants of the scheme; and hence, given
previous members of the senes we can predict the next member except
for the random element. The latter, though we may estimate its
variance, is itself unpredictable and there is thus an essential element
of uncertainty in any forecast of the future.

SUMMARY

1. Randomness in an oscillatory time series may conveniently be
tested by ascertaining the number of turning points which, in a random
series of n terms, has a mean value of | (^—2) and a variance of (lOw—
29) /90.

2. Alternatively, a test may be made of the first serial correlation which
has a variance of 1 /(w— 1) in random series.

3. The coefiicient of product-moment correlation between members of
a series (ft— 1) members apart is called the autocorrelation (for infinite
series) or the serial correlation (for observed series) of order ft.

4. The graph of the serial correlation as ordinate against the order ft
as abscissa is called the correlogram of the series.

5. For series which may be regarded as composed of a series of harmonic
terms, a technique known as periodogram analysis may be used to isolate
the periodic terms.

6. A series in which the value at any point is a function of values at
previous points plus a disturbance is said to be autoregressive ; and if
the function is linear is linearly autoregressive. The two most important
cases are —

7. The correlogram offers a means of descriminating between the
harmonic series and the autoregressive series,

8. An autoregressive series has no period in the strict sense. The
mean-distance between peaks may be quite different from the period of
the correlogram.

TIME - SERIES — (2)

EXERCISES

659

27.1 The following table shows the deviations from a moving nine-year
average of potato yields in England and Wales for the years 1888-1935
(units are -fe-th ton) —

Year

Yield

Year

Yield

Year

Yield

Year

Yield

1888

- 6

1900

- 7

1912

-15

1924

_ 1

4- 2

4- 6

4- 3

4 2

- 4

-- 3

4- 2

- 9

- 3

- 7

4- 1

- 3

- 1

04 1

4- 2

- 2

4 9

_j_ 0

4- 5

4 5

~ 2

+ 1

4- 4

4 1

4- 7

- 7

- 4

-10

4- 3

4- 8

- 3

4- 1

- 6

4“ 4

- 9

4- 2

+ 2

4- 3

4-11

4" 5

4* 4

- 1

- 4

Find the number of turning points and show that it does not differ
significantly from what would be expected of a random series.

27.2 From (27.35) derive an expression for the mean-distance between
peaks in a series of type (27.16) in the form

27r/cos-i{|(/^— 1)}

Consider the case when /a=0.

27.3 In an autoregressive series of type (27.17) find the mean-distances
between peaks for the following values of a and

- 1-5

0-8

- 1-0

0-6

- 0-8

0-8

Find also the autoregressive periods.

27.4 Show that the ^th auto-correlation of the first difference of a
series with autocorrelations pj^ is given by

27.5 In a series of type (27.17) the observed was 0*850 and tbe
observed fg =^0*606. Estimate a and /?.

66o

THEORY OF STATISTICS

27.6 Two series u and u' are added together so that a new series is formed

by If u and u’ are independent show that the /%th auto-

correlation of V is given by

Py. var var u'

wdiXu + var^"

where p and p' refer to the autocorrelations of u and u' respectively.

27.7 By considering the joint variation of and show that the
mean-distance between upcrosses in a series is 27r/cos”Vi where p^ is the
first autocorrelation. Find the expression in terms of a and § for series
of type (27.17).

27.8 The following are the serial correlations of the Beveridge series
referred to in Example 27.3. Draw the correlogram and compare any
periods which it suggests to you with the results of that example.

Order of
correla-
tion k

0-562

0*158

0*060

-0*036

0-103

0*109

-0*008

-0*013

-0*075

0*002

-0*039

0*042

-0 092

-0 075

0*007

0*062

-0-082

-0*062

0*056

0*065

-0-136

-0-021

0*010

0*050

-0-211

-0-062

-0-004

0*009

-0*261

-0*088

38 i

-0-015

-0*027

-0*192

-0*084

-0-047

-0*053

-0*070

-0-076

40 i

-0*047

-0 073

-0*003

-0-091

0*008

-0-106

12 !

-0*015

-0*052

0*034

-0 084

13 1

-0*012

-0*032

0*065

-0 019

14 ;

0*047

-0*012

0 099

0 003

15 1

0*101

0*059

0*009

0-010

27.9 For the autoregressive series of type (27.17) show that

l-Pi =

and hence that l+a4-/5 is not negative.

Show that the variance of the mean of n
series is

var u
n

(1+A)

consecutive terms of the

where, for large n, ^ is given by

_ 2(a+J?+P)

(1+/?) (l-f-a+y?)

TIME - SERIFS — (2)

66i

Hence show that A is negative if Pi is less than p, and thus that in some
circumstances the mean of n consecutive values can have a smaller variance
than the mean of n values chosen at random.

27.10 A Spencer 21-point smoothing formula (26.20) is applied to a
random series. Find the autocorrelations of the resulting series and
sketch the correlogram.

27.11 In the autoregressive series of type (27.17) consider the case when
/?=1. Show that the series then becomes undamped and the correlogram
reduces to a simple harmonic.

APPENDIX

663

APPENDIX TABLE 1
Normal curve

Ordinates of the Normal Curve with First and Second Differences

AM-)

0-0

0-39894

199

- 392

2-5

0 01753

395

4 79

0-1

•39695

591

~ 374

2-6

•01358

316

4 66

0*2

*39104

965

- 347

2-7

•01042

250

4 53

0-3

•38139

1312

- 308

2-8

00792

197

4 45

0-4

•36827

1620

- 265

2-9

•00595

152

4 36

0-5

•35207

1885

- 212

•00443

116

4 27

0*6

•33322

2097

- 159

3 1

•00327

4 23

0*7

•31225

2256

- 104

3-2

•00238

+ 17

0-8

•28969

2360

- 52

3 3

•00172

+ 13

0-9 .

•26609

2412

3-4

•00123

+ 10

1-0

•24197

2412

+ 46

3-5

•00087

4- 7

•21785

2366

+ 84

3-6

•00061

4 6

1-2

•19419

2282

+ 118

3-7

•00042

+ 4

1*3

•17137

2164 1

d- 143

3-8

•00029

1-4

•14973

2021

+ 161

3-9

•00020

4- 3

•12952

1860

4- 173

4-0

•00013

1-6

•11092

1687

+ 177

4-1

•00009

1-7

•09405

1510

4- 177

4-2

*00006

—

1-8

•07895

1333

4- 170

4-3

•00004

—

1-9

•06562

1163

4- 162

4.4

•00002

—

2*0

•05399

1001

4- 150

4*5

•00002

—

2-1

•04398 ^

851 i

+ 137

4-6

•00001

—

2 2

•03547

714

+ 120

4-7

•00001

—

2-3

•02833

594

+ 108

4-8

•00000

—

2 4

•02239

486

+ 91

Precision of Interpolation » — Owing to the magnitude of the second differences,
simple interpolation near the beginnmg of the table may give an error up to 5 m the
fourth place ; the use of second differences will bring this down to 1 or 2 in the last place,
third differences being small. Where third differences are greatest, in the neighbourhood
of ^/a'=0*6, the error may be as large as 3 m the last place unless the third difference
is used.

664

THEORY OF STATISTICS

APPENDIX TABLE 2

Areas under the normal curve (Probability function of the normal distribution)

The table shows the area of the curve j g lying to the left of

specified deviates x ; e.g. the area corresponding to a deviate 1 86 (=1 ‘S-f 0*36^
IS 0*9686.

Deviate

0*0 +

0*5 +

1-0 +

1-5+

2-0+

2*5 +

3*0 +

3*5 +

0*00

5000

6915

8413

9332

9772

92379

9*865

9*77

0*01

5040

6950

8438

9345

9778

92396

9*869

9*78

0*02

5080

6985

8461

9357

9783

9‘“413

9*874

9*78

0*03

5120

7019

8485

9370

9788

92430

9*878

9*79

0*04

5160

7054

8508

9382

9793

92446

9*882

9*80

0*05

5199

7088

8531

9394

9798

9246 I

9*886

9*81

0-06

5239

7123

8554

9406

9803

92477

9*889

9*81

0*07

5279

7157

8577

9418

9808

9M92

9*893

9*82

0*08

5319

7190

8599

9429

9812

925 O 6

9*897

9*83

0*09

5359

7224

8621

9441

9817

92520

9*900

9*83

0*10

5398

7257

8643

9452

9821

92534

9*03

9*84

0*11

5438

7291

8665

9463

9826

92547

9*06

9*85

0-12

5478

7324

8686

9474

9830

9256 O

ono

9>85

0-13

5517

7357

8708

9484

9834

92573

9*13

9*86

0*14

5557

7389

8729

9495

9838

92535

9*16

9*86

0*15

5596

7422

8749

9505

9842

92598

9*18

9*87

0-16

5636

7454

8770

9515

9846

92609

9*21

9*87

0-17

5675

7486

8790

9525

9850

92621

9*24

9*88

0*18

5714

7517

8810

9535

9854

02632

9*26

9*88

0-19

5753

7549

8830

9545

9857

02643

9*29

9*89

0*20

5793

7580

8849

9554

9861

02653

9*31

9*89

0*21

5832

7611

8869

9564

9864

02664

9*34

9*90

0-22

5871

7642

8888

9573

9868

92674

9*36

9*90

0*23

5910

7673

8907

9582

9871

02683

9*38

9*04

0*24

5948

7704

8925

9591

9875

02693

9*40

9*08

0*25

5987

7738

8944

9599

9878

92702

9*42

9*12

0*26

6026

7764

8962

9608

9881

92711

9*44

9*15

0*27

6064

7794

8980

9616

9884

92720

9*46

9*18

0 28

6103

7823

8997

9625

9887

92728

9*48

9*22

0*29

6141

7852

9015

9633

9890

92736

9*50

9*25

0*30

6179

7881

9032

9641

9893

92744

9*52

9*28

0*31

6217

7910

9049

9649

9896

92752

9*53

9*31

0*32

6255

7939

9066

9656

9898

9276O

9*55

9*33

0*33

6293

7967

9082

9664

9901

02767

9*57

9*36

0*34

6331

7995

9099

9671

9904

92774

9*58

9*39

0*35

6368

8023

9115

9678

9906

9278I

9*60

9*41

0*36

6406

8051

9131

9686

9909

92788

9*61

9*43

0*37

6443

8078

9147

9693

9911

92795

9*62

9*46 ;

0*38

6480

8106

9162

9699

9913

92801

9*64

9*48 !

0*39

6517

8133

9177

9706

9916

9*807

9*65

9*50

0*40

6554

8159

9192

9713

9918

02813

9*66

9*52

0*41

6591

8186

9207

9719

9920

02819

9*68

9*54

0*42

6628

8212

9222

9726

9922

02825

9*69

9*56

0*43

6664

8238

9236

9732

9925

92831

9*70

9*58

0*44

6700

8264

9251

9738

9927

92836

9*71

9*59

0*45

6736

8289

9265

9744

9929

02841

9*72

9*61

0*46

6772

8315

9279

9750

9931

92846

9*73

9*63

0*47

6808

8340

9292

9756

9932

02851

9*74

9*64

0*48

0844

8365

9306

9761

9934

9*856

9*75

9*66

^ 0*49

6879

8389

9319

9767

9936

9*861

9*76

9*67

Note : — Decsimal points in the body of the table are omitted. Repeated 9's axe
indicated by powets, e.g. 9*71 stands for 0*99971.

APPENDIX TABLE 3
Significance points of

Reproduced from Table III of R. A. Fisher’s StaUstical Methods for Research Workers, Oliver & Boyd, Ltd , Edmburgh, by permission of the author and publishers

APPENDIX

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^ r«< i-< T-< r-> i-< ■'-< T—t *—*

X ^

Sox05<M^'«tCMCM05a5XXC0iO'<fvr5CDE^r;jir)OC0d^05l£5^;JC0

^'^XCMW5XCDXXXOI>COXXT^iCOXCOrHOXg5C^O^^tNO

OOrMrt^I>i--<XOiOOCO'-^C>*X05«?CMXir><M05CpCM05CO'^'^COiOCO

oooocDrHT-icsiciicox-^-«Px»jr5CC)i>r''X<35a50’-i'^C'iX'«5i<Tr«u5^

^oxi>-^ciO'coxxx— 'r^oo 5 Cixxxot>cicocD;;^oog)iocox

5 rdT-i 05 «Or''X- 3 ’Xi/:'Ol''C'XCM — O'^XC 0 a 5 ’>!t‘ 05 U 5 C 105 ^C 0 l 0 ^
OOr-tCl CXM'OOXCrtO-^XCMXrfOCCCMXX-^XWOi-^XiOCiai

000 C 50 Qr-^^CM<MXX«^-

. E'* 00 X 05 O O r

<CMX^tOCpI>XCbO^<MX'«ttOXt'^X 05 0 ^ClCO; 5 ;«r 5 COt^OOXO

< 1 — iCMNdC^ddMClCMClX

N^0^.*~“For values of V greater than 30 the quantity -^^2^ may be taken to be distributed normally about mean -\/(2v 1) with

unit variance.

666

THEORY OF STATISTICS

APPENDIX

t-Table. — The Prop'Dition of the Area of the Curve of Unit Area lying to

i(v+i)

0 to 6, and for values

(Condensed to tnree figures trom the four-figure tables by “ Student " m Mttron, \o\ 5, 1925, and published

APPENDIX

667

TABLE 4

the Left ot tne Ordinate of Deviation t, for values of t proceeding by intervals of 0* 1 from

of V from 1 to 20

by permission of Meiron and the late W S Cosset, wno supplied a few corrections to the original tables)

^ I

0 500

•5000

0 500

0-500

0*500

0 500

0*1

•539

539

-539

•539

539

•539

539

•577

•578

578

•578

578

•578

-578

•578

578

* 3

•(315

*615

•615*

•616

616

•616

616

•616

-616

616

•4

•652

652

b52

•653

653

•653

• 5

686*

687

688

•688

•689

689

•6

720

•720

721

721*

722

•722

•7

• 751

•751

752

•753

753

754

-754

754

•8

•780

780

781

781*

782

•783

•9

•806

807

•808

808

•809

809

•810

811

1-0

•831

•831*

832

•833

833

•834

•835

835

1 1

853

853*

854

•855

856

857

•857

•857*

858

1 2

•872

873

•874

•875

876

•876

•877

•878

878

1-3

890

891

•892

•893

893

•894

•894*

•895

•896

1*4

•905*

•907

•907*

•908

•909

•910

•911

912

1-5

•919

920

•921

922

•923

923*

•924

•924*

•925

925

1-6

•931

932

•933

934

935

•936

•936*

•937

937

1-7

•941

•943

•943*

•944

945

946

•946

•947

•948

1-8

950

•951*

•952*

•953

•954

•955

•956

956*

1*9

958

959

9(30

•961

•962

962

•963

•964

964

2-0

■965

•966

967

'967

968

969

*970

970

•970

2-1

•970

•971

•972

•973

•973*

•974

•974*

975

976

2 2

•975

976

977

•978

•979

•980

980

2 3

•979

•980

•981

981

•982

982

■983

•983

•983*

•984

2-4

•982

•983

•984

•985

985*

•986

•987

2 5

■985

•986

•987

987

•988

•988*

•989

989

•989

2-6 1

•988

989

989*

•990

990

•991

991

2 7

•990

991

•991

•992

•993

993

•993

2 8

•991

•992

•992*

993

•993

•994

•991

•994

994^

2 9

993

•994

994

•991*

994*

•995

•996

3-0

•994

994*

•995

995

995*

•996

996*

3 1

•995

•996

•997

3 2

•996

•996*

•997

997

•997

•997*

■998

•998

3 3

996*

•997

997

•997

998

•998

3 4

•997

998

•998

•998*

999

3*5

•997*

•998

•998*

•999

3-6

998

•998

999

•999

■999

999

3 7

•998

•998*

•999

'999

•999

3-8

•998*

•999

999

3*9

•999

999*

•999*

1 000

4*0

•999

999

•999

•999*

1 000

4*1

•999

999

•999*

999*

1-000

4*2

•999

•999*

1 000

1-000

4*3

•999

•999*

1-000

4*4

. 999 *

1-000

4*5

•999*

4*6

1-000

Note , — ^The significance points of t for values of v greater than 20 can be derived by
taking the square-root of F (Table 5) for bearing in mmd that an x per

cent point of h ».orresponds to a value of 1 — J;r/I00 in the above table. Tn the above
table a small terminal ® means that the original four-figuie tables from which these
were compiled ended in a 5.

668

THEORY OF STATISTICS

APPENDIX TABLE 5 — Significance points of the variance-ratio F

A. 5 per cent points

Reproduced from Fisher and Yates Statistical Tables for Biological, Medical and Agricultural Research,
Oliver and Boyd Ltd , Edinburgh, by permission of the authors and publishers

161

•4

199

•5

215

•7

224

•6

230

234

•0

238

•9

243

•9

249

254

■3

•51

•00

•16

•37

•41

•50

•13

•55

•12

•84

•74

•53

•71

•94

-59

•39

•16

•04

•91

•77

•63

•61

•41

•19

•68

•53

•36

•99

•14

•76

•84

•67

•59

•35

•12

•97

•87

•57

•23

•32

•46

•07

•84

•58

•44

•28

•12

•93

•12

•86

•63

■48

•23

*71

■96

■10

■71

■48

•22

•07

■91

•74

•54

■84

3 -

■36

■09

•95

•61

•40

■75

■49

•26

■11

•00

•69

•30

3 '

■41

■18

■02

•77

•60

•42

•21

3 '

•11

•85

•53

•35

•13

3 '

■29

•79

•29

•74

•59

2 '

■42

■24

■70

•55

■96

•93

■77

•66

•51

•34

■92 1

•90

•74

•63

■48

■31

■11

■10

■87

•60

•08

!•

■84

■07

2 ’

■68

■57

■42

■05

■55

■03

•80

■64

■53

■38

■40

■01

•62

■99

'76

■60

■49

2‘

■22

■98

■74

■59

!•

■21

■96

: 28

•20

■34

■95

!•

1 29

•18

■33

2 '

•93

!•

■17

■32

•92

■69

•08

■23

■84

■61

•00

■15

•76

120

•92

•07

■68

■45

!■

•84

2 ‘

•99

•60

!•

Lower 5 per cenir'points are found by interchange of pj and v^, i.e always

correspond with the greater mean square

APPENDIX

669

APPENDIX TABLE 5 — {continued ) — Significance points of the variance-ratio F

B. 1 per cent points

Reproduced from Fisher and Yates Sfahsticul Tabhs for Btologtcal, Medical and Research,

Oliver and Boyd Ltd , Edinburgh, by permission of the authors and r-u

^1\

4052

4999

5403

5625

5764

5859

5981

6106

6234

6366

98 49

99 00

99-17

99 25

99-30

99-33

99-36

99 42

99-46

99-50

34-12

30.81

29-46

28-71

28-24

27-91

27*49

27*05

26-60

26-12

21-20

18*00

16-69

15-98

15*52

15-21

14 80

14*37

13-93

13-46

16-26

13*27

12*06

11 39

10*97

10 67

10*27

9-89

9-47

9 02

13 74

10-92

9-78

9-15

8-75

8-47

8-10

7*72

7-31

6-88

12-25

9-55

8 45

7-85

7*46

7-19

6 84

6-47

6-07

5-65

11-26

8*65

7*59

7-01

6*63

6-37

6-03

5-67

5-28

4 86

10-56

8-02

6-99

6*42

6*06

5-80

5-47

5-11

4*73

4 31

10 1

10-04

7-56

6-55

5-99

5-64

5-39

5*06

4-71

4-33

3-91

9-65

7-20

6*22

5 67

5-32

5-07

4*74

4*40

4-02

3-60

9 33

6*93

5-95

5*41

5-06

4-82

4*50

4*16

3-78

3*36

9-07

6*70

5-74

5*20

4*86

4*62

4-30

3*96

3-59

316

8-86

6-51

5-56

5-03

4-69

4-46

4*14

3-80

3-43

3-00

8 68

6-36

5-42

4*89

4-56

4-32

4*00

3-67

3-29

2-87

8-53

6*23

5*29

4*77

4*44

4-20

3*89

3*55

3-18

2-75

8-40

6*11

5*18

4-67

4*34

4-10

3*79

3*45

3-08

2-65

8-28

6*01

5 09

4*58

4*25

4-01

3-71

3*37

3-00

2-57

8*18

5-93

5-01

4*50

4*17

3-94

3*63

3-30

2-92

2-49

8-10

5*85

4*94

4*43

4*10

3-87

3*56

3*23

2*86

2-42

8-02

5*78

4-87

4*37

4*04

3-81

3*51

3*17

2-80

2-36

7-94

5*72

4*82

4*31

3*99

3*76

3*45

3*12

2-75

2-31

7-88

5*66

4*76

4*26

3-94

3-71

3*41

3-07

2-70

2-26

7-82

5*61

4-72

4*22

3-90

3*67

3*36

3*03

2*66

2-21

7-77

5*57

4-68

4*18

3*86

3*63

3*32

2*99

2*62

2*17

7-72

5*53

4*64

4*14

3-82

3*59

3*29

2*96

2-58

2-13

7*68

5*49

4*60

4*11

3-78

3-56

3-26

2*93

2*55

2-10

7-64

5*45

4*57

4*07

3-75

3-53

3*23

2*90

2-52

2-06

7-60

5*42

4*54

4*04

3*73

3-50

3*20

2-87

2-49

2*03

7-56

5-39

4-51

4-02

3*70

3*47

3-17

2 84

2-47

2*01

7*31

5*18

4-31

3*83

3*51

3-29

2*99

2*66

2-29

1*80

7*08

4*98

4*13

3*65

3-34

3*12

2-82

2-50

2-12

1*80

120

6*85

4*79

3*95

3*48

3*17

2 96

2-66

2 34

1-95

1*38

6*64

'4-60

3-78

3-32

3-02

2*80

2-51

2-18

1-79

1*00

Lower 1 per cent points are found by interchange of and Vg. i.e. must always
correspond with the greater mean square

670

THEORY OF STATISTICS

APPENDIX TABLE 5 — (continued) — Significance points ot the variance-ratio F

C. 0*1 per cent points

Reprofiuced from Fisher and Yates. Staiistual Tables for Biological, Medical and Agricultural Research,
Oliver and Boyd Ltd., Edinburgh, by permission of the authors and publishers

V-
* \

405284

500000

540379

562500

576405

585937

598144 610667

623497

636619

998-5

999*0

999-2

999*2

999-3

999-4

999*4

999 S

999*5

167-5

148*5

141 1

137*1

134 6

132-8

130*6

128*3

125 9

123-5

74-14

61*25

56*18

53*44

51-71

50-53

49*00

47*41

45-77

44-05

47-04

36*61

33*20

31-09

29 75

28*84

27-64

26-42

25-14

23-78

35 51

27-00

23*70

21-90

20-81

20-03

19*03

17-99

16-89

15-75

29 22

21*69

18*77

17 19

16-21

15-52

14 63

13-71

12-73

11-69

25-42

18*49

15 83

14*39

13 49

12-86

12-04

11-19

10-30

9-34

22-86

16*39

13 90

12 56

11-71

11-13

10*37

9-57

8-72

7-81

21-04

14*91

12*55

11 28

10-48

9-92

9*20

8-45

7-64

6-76

19*69

13*81

11*56

10 35

9-58

9-05

8 35

7-63

6-85

6-00

18*64

12 97

10*80

9*63

8 89

8-38

7*71

7-00

6-25

5-42

17*81

12*31

10 21

9 07

8-35

7-86

7-21

6-52

5-78

4-97

17-14

11*78

9*73

8*62

7-92

7-43

6*80

6-13

5-41

4-60

16*59

11*34

9*34

8*25

7-57

7-09

6-47

5*81

5-10

4-31

16*12

10*97

9*00

7*94

7-27

6-81

6-19

5-55

4-85

4-06

15*72

10*66

8*73

7*68

7-02

6-56

5*96

5-32

4*63

3-85

15-38

10*39

8*49

7*46

6-81

6-35

5 76

5 13

4-45

3-67

15*08

10 16

8*28

7*26

6-61

6-18

5*59

4-97

4-29

3-52

14*82

9*95

8*10

7-10

6-46

6-02

5 44

4-82

4*15

3-38

14*59

9*77

7*94

6-95

6-32

5-88

5*31

4-70

4*03

3-26

14-38

9*61

7*80

6*81

6-19

5-76

5*19

4-58

3-92

3-15

14-19

9*47

7*67

6-69

6-08

5*65

5 09

4-48

3-82

3-05

14-03

9*34

7*55

6-59

5-98

5-55

4*99

4*39

3-74

2-97

13-88

9*22

7*45

6-49

5-88

5-46

4-91

4-31

3-66

2-89

13-74

9*12

7-36

6*41

5-80

5-38

4-83

- 4-24

3-59

2-82

13*61

9*02

7*27

6-33

5-73

5-31

4-76

4-17

3-52

2-75

13*50

8*93

.7*19

6-25

5-66

5-24

4-69

4-11

3-46

2-70

13-39

8-85

7*12

6-19

5-59

5-18

4-64

4-05

3-41

2-64

13*29

8*77

7*05

6*12

5-53

5*12

4-58

4-00

3-36

2-59

12*61

8-25

6-60

5-70

5*13

4-73

4*21

3-64

3-01

2-23

11*97

7-76

6*17

5-31

4-76

4-37

3-87

3*31

2-69

1-90

120

11*38

7-31

5*79

4-95

4-42

4 04

3-55

3-02

2*40

1*56

10 83

6-91

5-42

4-62

4-10

3-74

3-27

2-74

2-13

1-00

Lower 0-1 per cent points are found by interchange of and must always

correspond with the greater mean, square

APPENDIX

671

APPENDIX TABLE 6. — Significance points of the distribution of z

A. 5 per cent points

Reproduced bv Kind permission of Professor R, A Fisher and Messrs Oliver and Boyd from the former's
Statistical Methods for Research Workers

2 5421 2

•6479 2

6870 2

7071 2

7194 2

7276 2

7380 2

7484 2

•7588 2

7693

1*4592 1

•4722 1

■4765 1

•4787 1

•4800 1

•4808 1

4819 1

4830 1

*4840 1

•4851

1*1577 1

*1284 1

•1137 1

•1051 1

•0994 1

*0953 1

*0899 1

•0842 1

•0781 1

■0716

1*0212

•9690

*9429

•9272

•9168

*9093

*8993

•8885

•8767

•8639

*9441

*8777

•8441

•8236

•8097

*7997

*7862

7714

*7550

•7368

8948

•8188

•7798

•7558

•7394

*7274

•7112

6931

*6729

•6499

•8606

•7777

•7347

*7080

•6896

•6761

6576

•6369

•6134

•5862

•8355

•7475

*7014

*6725

*6525

•6378

*6175

•5945

5682

•5371

•8163

*7242

•6757

•6450

•6238

•6080

•5862

•5613

•5324

•4979

•8012

•7058

•6553

*6232

•6009

•5843

•5611

•5346

•5035

•4657

•7889

*6909

*6387

•6055

•5822

•5648

•5406

•5126

•4795

•4387

•7788

•6786

•6250

•5907

•5666

•5487

*5234

•4941

•4592

•4156

*7703

*6682

•6134

•5783

•5535

•5350

•5089

•4785

•4419

•3957

•7630

•6594

•6036

•5677

•5423

•5233

•4964

•4649

•4269

•3782

•7568

•6518

•5950

•5585

•5326

•5131

•4855

•4532

•4138

•3628

•7514

•6451

•5876

•5505

•5241

•5042

•4760

•4428

•4022

•3490

•7466

•6393

•5811

•5434

•5166

•4964

•4676

•4337

•3919

•3366

•7424

•6341

•5753

•5371

•5099

•4894

•4602

•4255

•3827

•3253

•7386

•6295

•5701

•5315

•5040

•4832

•4535

•4182

•3743

•3151

•7352

•6254

•5654

•5265

•4986

•4776

•4474

•4116

•3668

•3057

*7322

•6216

•5612

•5219

•4938

•4725

•4420

•4055

•3599

•2971

•7294

•6182

•5574

•5178

•4894

•4679

•4370

•4001

•3536

•2892

•7269

•6151

•5540

■5140

•4854

•4636

•4325

•3950

•3478

•2818

•'7246

•6123

•5508

•5106

•4817

•4598

•4283

•3904

•3425

•2749

•7225

•6097

•5478

•5074

•4783

•4562

•4244

•3862

•3376

•2685

•7205

•6073

•5451

•5045

•4752

•4529

•4209

•3823

•3330

•2625

•7187

•6051

•5427

•5017

•4723

•4499

4176

•3786

•3287

•2569

•7171

•6030

•5403

•4992

•4696

•4471

•4146

•3752

-3248

•2516

•7155

•6011

•5382

•4969

•4671

•4444

•4117

•3720

•3211

•2466

•7141

•5994

•5362

•4947

•4648

•4420

•4090

•3691

•3176

•2419

•6933

•5738

•5073

•4632

•4311

•4064

•3702

•3255

•2654

•1644

•6729

■5486

•4787

•4319

•3974

•3706

•3309

•2804

•2085

0-000

672

THEORY OF STATISTICS

APPENDIX TABLE 6 — (contd ,) — Significance points of the distribution of z

B. 1 per cent points

Reproduced by kmd permission of Professor R A Fisher and Messrs Ohver and Boyd from the former’s
StaiisUcal Methods for Research Workers

APPENDIX

673

APPENDIX TABLE 6 — {confd .) — Significance points of the distribution of z

C 0*1 per cent points

Reproduced by kind permission of Professor R A Fisher, Dr W E Deir.rg ord Oliver and Boyd

from Prof. Fisher’s Statistical Methods for Hc'-eir'’, V, or/: ’rs

ANSWERS TO THE EXERCISES

AND HINTS ON THEIR SOLUTION

CHAPTER 1

26,287

{AB)

887

2,308

{AC)

374

(B)

2,853

{BC)

353

(C)

749

{ABC)

149

(ABC)

156

{aBC)

179

(ABy)

431

{aBy)

1,249

(A/3C)

272

{apC)

163

{Ah)

759

(c^h)

20,504

1 3. The frequencies not given in the question itself are —

(a) (AB) 107 (AC) 405 (BC) 525.

(b) (AJy) 22,980 (aBy) 13,585 (a^C) 96,478 (cc/Sy) 28,868,495.

1 4.

(AB) {B)

{AP)^<J)

(AB) ^ (B)

•• {AB) + {A^)^ {B) + {fi)

that is

(AB) (A)
'{B) ^ N'

that 1 -

{B)-{AB)^ N-{A)

that IS

(AB) (A)
(coB) ^ (a)

1.7. 160 Take .4 —husband exceeding wife in first measurement, ^ — husband
exceeding wife in second measurement, and find (acj9),

1.8 38 If A, B, C denote passing first, second and third examinations, (C),
(a^C) and (A By) are all that is necessary to answer the question. The other five
frequencies (including N) are redundant.

Further, N~ (ccfiC) '-{ccjSy}^(A}-i-(B)~-(ABC) — (ABy), i.e there is a linear
relation between the given frequencies and the ultimate frequencies are therefore
indeterminate.

1.9. 10 per cent.

1.11. Denoting government, voting for the motion and English membership by
4, F, C, we have (ABC)^300, (a5C)=53, (AfiC)^lO, {ay^C)==102, (ABy)^3Q
(aFy)=72, (A^y)^8, (a/?r)«25.

1 13. 80/263 or 304 per thousand.

1.14. 55/85 or 65 per cent.

1.15. 32 per cent and 30 per cent.

1.16. 117.

1.17. 108.

1 . 20 . {1— (1-1-23'), i.e. p must lie between 0 and i (1—23') or between
i (1+23) and J.

1.21, As a hint, remember the condition that —

iBC)-^(B) + (C)-N

1.22 HA, B, C denote liking chocolates, toSee or boiled sweets, (a^y) is negative.

675

676

THEORY OF STATISTICS

CHAPTER 2

2,1 Deaf-mutes from childhood per million among males 222 , among females
183, there is therefore positive association between deaf- mutism and male sex , if
there had been no association between deaf-mutism and sex, there would have been
3,176 male and 3,393 female deaf-mutes

2.2. {a) Positive association, since 1,457.

(b) Negative association, since 294/490=3/5, 380/570=2/3.

2.3 Percentage of Plants above the Average Height

Parentage Crossed Self-fertilised

. 86 per cent - 25 per cent

79 „ 17

78 „ 34

71 45

50 „ 35

Tpomsea purpurea
Petunia violacia
Reseda iutea
Reseda odorata .
Lobelia fulgens

The association is much less for the species at the end than for those at the beginning
of the list.

2 4. Percentage of dark eyed amongst the sons of dark-eyed fathers 39 per cent.

Percentage of dark-eyed amongst the sons of not dark-eyed fathers 10 per cent

If there had been no heredity, the frequencies to the nearest unit would have been
{AB)o 18, {Afi)o 111, (aB)o 121, (ay5)o 750.

2 5. Percentage of light-eyed amongst the wives of light-eyed husbands 59 per cent.

Percentage of light-eyed amongst the wives of not light-eyed husbands 53 per cent.

If there had been no association : (AB)qc=298, (Aj^)o=: 225, (aB)o=143, (a^)o=108.

2 6. The following are the proportions of the insane per thousand in successive
age-groups —

In general population : 0*9, 2*3, 4*1, 5*7, 6*9, 7*5, 7*7, 6*8
Amongst the blind: 20*1, 16-0. 16*3, 20*7. 18*3, 17-8, 11-4, 5*3

Note the diminishing association, which is especially clear in the age-group 65-,
and the negative association in the last age-group. The association coefficient gives
the values below, which decrease continuously —

Association coefficient : -{-0-92, -j-0*75, 4-0*61, 4-0 57, 4-0*46, 4-0*41, 4-0*20,

-0*13.

2 10. 4-0*90.

2 11 4-0*70.

2 13. The frequencies are, for association —

2.14.

(1)

{AB)

{xB)

(2) •

(AB)

[cip)

(3)

(AB)

disassociation —

(1)

{xB)

(2)

(AB)

(xB)

(3)

(xB)

(D)/N =6*9

per cent

{A)/N = 6*8

(AD}/(A) =45*0

{AD)/[D) =44*6

=3*6

.(AP)/(P) = 4*7

(A/?I))/{A/^} =41*2

{ApD)l{pB) =54*9

(BD)/{B) <^42-7

{AB)I{B) =29*2

(ABD}/(AB) =51*6

(ABD)/(BD) =35-3

)ve give two legitimate comparisons

The general results

for the boys, i.e. a very small association between development defects and dulness

ANSWERS TO THE EXERCISES

677

amongst those exhibiting nerve signs, as compared with those who do not exhibit
nerve signs, or with the girls m general. As the association amongst those who do
not exhibit nerve signs is quite as high as for the girls m general, the “ conclusion
quoted does not seem valid

15.

(1)

(2)

(1)

(2)

Per

thousand

(B)/N

3*2

7*5

(A)/N

0*9

4*0

(AB)/(A)

14*9

11*7

{AB)/{B)

4*0

6*3

(BC)/{C)

38*8

63*0

{AC)/{C)

6*6

18*8

{ABC}/ {AC)

216

214

{ABC)/{BC)

36*8

63*8

The above give the two simplest comparisons, either of which is sufficient to show
that there is a high association between blindness and mental derangement amongst
the deaf-mutes as well as association m the general population ; amongst the old,
the association is, in fact, small for the general population, but well-marked for deaf-
mutes. This result stands in direct contrast with that of Exercise 2 14, where the
association between the two defects A and D was much smaller in the defective popula-
tion /? than m the population at large As previously stated, no great reliance can be
placed on the census data as to these infirmities

2.16. If the cancer death-rates for farmers over 45 and under 45 respectively were
the same as for the population at large, the rate for all farmers over 16 would be
2*726. This IS slightly greater than the actual value 2*633 but the difference would
not justify any statement that farmers were peculiarly liable to cancer," or not.

2 17. 15 per cent.

2.19 If ^ and B were independent in both C and y populations, we should have
(yl£) equalto 471 x 419 , 151 x 139 _ „ . -

617 ' ■ 383 ■

Actually [AB) is only 358. Therefore A and B must be disassociated in one partial
population or both

2.22. (1) 68*1 per cent (2) 42*5 per cent The possible fallacy that a total
association between " spending more than one’s opponent " and " winning " only
meant that Conservatives spent more and that Conservative principles carried the
day IS now avoided, and there seems no reason for declining to consider this as evidence
of the effect of expenditure on election results

2 23 The limits to y are y<l{3x-x^-l)

>i(x+x^)

subject to the conditions y^O, y^2Ar— 1. No inference of a positive association
from two negatives is possible unless Af lies between the limits 0 382 . . , 0*618 . .

2 24. The limits to y are

(1) y<i(6Ar-6Ar*-l)

>i{Af-f6;r^)

subject to conditions y^O,

An inference is only possible from positive associations of A B and AC if at ^ J ,
an inference is only possible from two negative associations if x lie between 0*211
and 0*274 . . . Note that x cannot exceed

(2) y<i{6;r-3Ar«-l)

>U2x-h3x^)

subject to conditions y^-0, ^Sat— 1, ^at.

No inference is possible from positive associations oi A B and BC
An inference is only possible from negative associations if x he between 0*183 .
and 0*215 .. . Note that x cannot exceed J.

(3) ' y<^(6Ar— 2Ar®~l)

>li3x+2x^)

subject to the conditions y^O, '^5x—t, ^x

As in (2), no inference is possible from positive associations oi AC aridr BC ; an
inference is possible from negative associations if x lie between 0*177 . and
0*224 . . . Note that x cannot exceed

678

THEORY OF STATISTICS

CHAPTER 3

31. A, 0-68; B, 0-36

3.2 C-=0*02, r*=o-oi

3.4. The table is not isotropic as it stands. It becomes positively so if the columns
are arranged in the order A^, A^, A^, A A and the rows in order (from top to bottom)
Bz, B^, Bj.

3.5. C«0-05, 7=0-03.

3.7. C=0*40. For a large number such as 1,000 this is probably significant, i e.
not due to fluctuations of sampling. From inspection of the tables the contingency
IS positive, i.e tnis evidence would suggest that persons tended on the whole to prefer
music of their own nationality. But Qiere are exceptions, e g. the English.

In any case these data are purely imaginary, and it is not suggested that they reflect
m any way the true state of affairs

3.8. C=0-23, T=0*17 suggestive of sbght association.

3.10, C=0-10

CHAPTER 4

4.1. 1200. 200.

4.2. 270, 40.

4.3. 92-375.

4.4. 216-5

4 5. (a) J -shaped ; (6) U-shaped; (c) single-humped moderately asymmetrical,
{d} J-shaped in all three cases

CHAPTER 5

5 2. 14-58.

5 3. Mean, 156-73 lb Median, 154-67 lb. Mode (approx.), 150-6 lb. (Note that
the mean and the median should be taken to a place of decimals further than is desired
for the mode ; the true mode, found by fitting a theoretical frequency curve, is 151 • 1 lb.)

5. 4. Mean, 0*6330, Median, 0*6391. Mode (approx.), 0-651. (True mode is
0-653.)

5.5. About ;^3,250.

5.6. Mean«^.

5.7. (I) 82*75, (2) 81-78, (3) 80-25, (4) 80-25.

5.8. Arithmetic means=— ^(2«+i — 1).

ti-p i

Geometric mean ««2*.

Harmonic mean

5.9. Mean=«^. If the terms of the given binomial series are multiplied by 0, I,
2, . . note that the resulting senes is also a binomial when a common factor is removed
(A full proof is given in Chapter 10.)

5.11, (1) 921.507, (2) 916,963.

5.12, For N.M. specials, 15s. Id. per 120 ; for ordinaries, 12$, 9d. per 120.

ANSWERS TO THE EXERCISES

679

CHAPTER 6

6.2. Standard deviation 21*3 lb. Mean deviation 16*4 lb Lower quartile 142*5,
upper quartile 168-4; whence g=12*95. Ratios m.d./s d =0*77, g/sds=0*61.

6.3. Median=;f3,250, upper quartile =;f5,000, 9th decile =;^8, 600 approximately

6.4 gi = 24*13 years. Median=27 29 years. 03=32*19 years 0=4*03 years.

6.5. 2*872

6 6. This proposition is equivalent to the one that the square of the mean of a set
of positive numbers is less than the mean of the squares. This is proved in most
textbooks on Algebra.

6 8 (1) Af=73*2, o-=17*3, (2) M=73-2, (7=17*5. (3) M = 73*2, cr=18*0.

(Note that while the ‘'mean is unaffected m the first place of decimals, the standard
deviation is higher the coarser the grouping )

6.9. England, (7=2*55, Scotland, (7=2*48, Wales, (7=2*33; Ireland, (7=2*15
inches. For the weight ciistribution (7=21*14 lb.

6.10. V fipq The proof is given in Chapter 8.

6.11 The assumption that observations are evenly distributed over the intervals
does not affect the sum of deviations, except for the interval m which the mean or
median lies , for that interval the sum is ^^(O 25-f-(f*), hence the entire correction is

d{n^ -W3) 4-W2(0* 25-f

In this expression d is, of course, expressed as a fraction of the class-interval, and is
given its proper sign.

6.14. 3*80, 3*65, 3*53, 3*20.

CHAPTER 7

7.1. In class-intervals of 1 0 lb.

/A2=4*470, /43=6*927, /(i4«89*119; /?i=0*537. /?3=4*461.

Curve leptokurtic.

7.2 0 06, 0-29, 0*27.

7.3. /^2=li*375, /(3=12*705, /i4=428*708, m class-intervals of 1 gallon.
yffi = 0*110, y^a=3*313

Measures of skewness are 0*027, 0*14, 0*15 The second is obtained by approxi-
mating to the mode in the manner of 5.26

7.4. Before corrections, /A3=7*301, /<3=0*166, /44=163*465;

After corrections, /(-2=6*551, /fc3=0*166, /^ 4 = 132*975

Note that the small negative /tg in the finer grouping becomes positive in the coarser
grouping.

7 . 5 . /i^^npqiq—p)

~ 3 ^^%^ + pqn { 1 — 6^^) .

7.6. About the mean, /<2=14*75, /i3=39*75, /A4=142*3125.

About the origin, /A2'= 21, /43'=166, /i4'=1132.

7.8. This proposition is equivalent to that of Exercise 6 6. For U-shaped popula-
tions y? 2 < 2 .

7 9. Ara=7*057, /C3=36*152, /C4=259*335.

CHAPTER 8

8.1. 27*31 per cent.

8.2. Expected frequencies are : 1, 12, 66, 220, 495, 792, 924, 792. 495, 220, 66, 12. 1.
Expected mean=6 ; expected cr= 1*732,

Actual mean =6*139; actual <r= 1*712.

68o

THEORY OF STATISTICS

4096

Expected frequencies, to nearest units, are 2, 11, 51, 178, 438, 765, 951, 841, 529,
236, 75, 17, 3, totalling 4097 , (these are obtained by simple interpolation in Appendix
Table 1).

8 4. 17,

8.5 If p is the expectation of getting an even number,

Hence, and the number of times is 1 0,000 once.

8-8. The frequency of r successes is greater than that of r—l so long as r<yip'^p,
if np IS an integer, r=^np gnes the greatest term and also the mean
8.9. This follows at once from a consideration of the Galton-Pearson apparatus

Binomial

Normal curve

1 7

10-5

42-7

120

116 1

210

211-5

252

258-4

210

211*5

etc.

8.11. Mean 74-3, standard deviation 3-23

8.12, About zero mean the deciles are ;
the corresponding negative values

8 . 13 .

2*57\/2;r

0, 0 2533, 0-5244, 0-8416, 1-2816, and

Calculated mean and quartile deviations, 2-05 and 1-73 (observed, 2-02 and 1-75)
These figures are in units of one inch

814. Calculated mean and quartile deviations (years), 6-37 and 5-38 (observed
5-44 and 4 03)

8,15, 18

816 o'=2-267 (uncorrected)

Theoretical frequencies, 2, 5, 11, 20, 29, 35, 35, etc.

8.17. Theoretical frequencies, 336-5, 397-1, 234-6, 92-5, 27-3, 6-5, 1*3, 0*2.

8.18. a: 8=1-362, fC3«1.766. fr4=2*510.

CHAPTER 9

9.1, cr^=l*414. o'^«2-280, y=+0*81.

Ar==:0-5y+o-5, y-=i*3A:4-M.

9-2 r (between X and y) = — 0-66; between Y and 2=0-60; between 2 and
A"«-~0-13.

9 4. f=-j-0-96

9.5. (1) -0-41, (2) +0-40.

CHAPTER 10

10 3. Erom equations (10.11) and (10 12) replace <7^ and <Ta by Sj and Sg in equation
(10.10) Regarding this as an equation for 7, note that is a maximum when tan
2$ is infinite, or ^=45*^.

10.4. In fig. 10 1 suppose every horizontaT array to be given a slide to the right
until its mean lies on the vertical axis through the mean of the whole distribution :

ANSWERS TO THE EXERCISES

68i

then suppose the ellipses to be squeezed m the direction of this vertical axis until
they become circles. The original quadrant has now become a Sector with an angle
between one and two nght angles, and the question is solved on determining its
magnitude.

10 5 The ellipse is a honzontal section of the surface Its equation is

and the standard deviations of sections are the square roots of the lengths of radii
vectors of the ellipse

10 6. The maximum and minimum s.d.^s are given by the principal axes, which
leads to equations (10.11) and (10.12).

For an intermediate value there are two radii vectors and hence two sections.

10 8 a and b must be negative, and ab—h^^O

^ab-h^'

■i

ab-h^

-Jl

V ah

10 9. The sum of the pth. powers of the first n natural numbers is plus

terms of lower order in n

10.10. Use equation (9.11),

CHAPTER 11

11.1. iy^=0-242, '7y*=0-266.

11.2. 17xir==0-82, l/yaj^O-SO.

11.3. p==+0-79.

11.4. If the judges be denoted by 1, 2, 3,

Piib=— 0‘21, p23«-- 0*30, Pi3*=-f0*64

This suggests that j’udges 1 and 3 have tastes in common, but neither has much
in common with judge 2.

11.5. 8=2/3.

11.6. 8=0*77.

11.10. y=+0*83.

11.11. ^=+0*22, 11,868 entries.

CHAPTER 12

12.1. fi2.a=+0-759, ^' 13 . 2 = + 0*097, i=-0*436.

*^j.2s=^‘ 04, £r2 j3=0*594, <73j2=70*l.

A'l = 9 • 3 1 + 3 • 37^2 + 0 • 00364 A' 3 .

12.2. Ri( 23)=0*80, i?2(3j}=0*84, R3(i2)=0*57

12.3. fi*.34= +0*680, ^13.34= +0-803, 4 . 23 = +0*397.

^38.14= — 0*433, ^84,13= "~0* 553, 0*149.

0-1.234=0*17, 0*2 j34=s 49*2, Og 134 = 12*5, 0*4 jgjess 105 *4.

A:i«53+0*127Ar2+0*587A'3+0*0345A’4. ,

12.4. i?i(28>=0*87, Ri(234)=0*89.

12.5. {.Sri-I9*9)=4*51(A'j™49*2)^0*88(Ar3

>'ik.s=--0‘03.

-30*2)

-0*072(;sf4-

.4814)+0*63(A'*-«41*6)

682

THEORY OF STATISTICS

12.7. Number of order s^n

Total number =^{2'*-"^-'l}

This mcludes coefHcients of type i?i(g) and counts as differert from

12.8. The correlation of the ^th order is r/(l+pr). Hence if r be negative, the
correlation of order w —2 cannot be numerically greater than unity and r cannot exceed
(numerically) 1/(m--1).

12.9. rjjj 3= — 1, ^23.1

12.10. ^12 3==»'is.2®=%.1=~'^*

CHAPTER 13

13.1. In Table 9.5 the unit, being a weekly figure, is not modifiable to the extent
that it relates to the situation at a given point of time. The choice of different intervals
between the points (e g, months) might, perhaps, give a somewhat different picture.

In Table 9.6 the unit is a registration distnct and is modifiable by the amalgamation
of districts.

13.2. For this series ye=— 0*87. This is to be regarded as a nonsense-correlation,
although a very profound analysis might suggest that the falling infantile mortality
was due to technical progress which also made increases in population possible.

13.3. During the period steam vessels were replaced by diesel oil burners to some
extent and horse-drawn vehicles by oil-propelled vehicles. From this point of view
the correlation is hardly nonsense, though the relationship is very remote.

CHAPTER 14

14.1. Estimated true standard deviation 6*91 ; standard deviation of fluctuations
of sampling 9 ’38. (The latter, which can be independently calculated, is too low,
and the former consequently probably too high. Cf. 17.30.)

14.2. 0-43.

14.3. 58 per cent.

14.4. cr,V -v/ (£r7+^)(<7'2*+^^^^

acTi

14 6. 0-29

The others may be wntten down from symmetry

14,8 (1) No effect at all. (2) If the mean value of the errors m variables is d, and
in the weights «?, the value found for th*e weighted mean is —

The true value -f d? — f

If r is small, d is the important term, and hence errors in the quantities are usually
of more importance than errors in the weights. If r become considerable, errors in
the weights may be of consequence, but it does not seem probable that the second
term would become the most important m practical cases.

14.9. +0*036.

var B

14.10. A+var B)(var B+var C)}

ANSWERS TO THE EXERCISES

683

CHAPTER 15

15.1. Line: y==2-58-f- M 3(^-2)

Quadratic : y= 1-48+ 1 * 13(Ar-2) 4'0-55(A~2)2

Cubic: y=l-48+0 025(A'-2) + 0-55(X-2)2+0-325(Ar-2)3

Sums of squares of residuals * 5*819, 1*584, 0*063

15 2. If y IS the average number of children for the duration X to A + l years —

Line: y=3-814 + 0-887(^^-3^

Quadratic. y=4-351+0 887(^'|-3 j-0- 134(^'|-3y
Cubic: y=4-351+0-912^'|-3^-0-134(^^-3y-0 00361

For X = 1 7 the three values are 4 • 1 7, 4 • 68, 4 * 69
15 3 7 c=1.42

15.4. A' = Gross output per 100 labour, y —gross output,
y = 48 • 33 4- 0 * 2375 A - 0 • 00005546 A ^

CHAPTER 17

17 1. Tbeo. Ar==6, cr«l*732: Actual ikr«6*116, cr=:l*732.

17.2. [a) Theo. Ar«2*5. cr=M18: Actual Af=2*48, (r=l*14

{b) „ iW=:3. a'==l*225: „ M«2*97, cr«l*26

17.3. The standard deviation of the proportion is 0*00179, and the actual divergence
is 5*4 times tliis, and theiefore almost certainly sigmficant.

17.4. The standard 'deviation of the number drawn is 32, and the actual difference
from expectation 18. There is no significance.

17.5. Difference from expectation 7*5 ; standard error 10*0. The difference might
therefore occur frequently as a fluctuation ot sampling

17.6. Standard error of proportion of bad eggs= 1*6536 per cent. A range of three
times this gives range of 7*5 per cent to 17*5 per cent approximately,

17.7. The test can be applied either by the formulae of Case 2 (17 28) or those of
Case 3 (17.29). Case 2 is taken as the simplest.

(^B)/(R)=70*1 per cent. ; (Aj3)/(/3)=^64-S per cent.

Difference 5*8 per cent. {A)/N^67-6 per cent and thence 3*40 per cent. The
actual difference is 1 * 7 times this and might, rather infrequently, occur as a fluctuation
of sampling,

17.9. Difference of proportions= ei2=^’033. Difference significant. Similar
conclusions follow if the formulae of Case 3 (17.29) are applied.

17.10. Proportion 36 per cent. Limits 32*4—39*6 per cent. The sampling is
almost certainly not simple. Possible causes are : (a) nature of subject-matter might
require words of certain type, e.g scientific words probably would not be Anglo-Saxon ;
{b) the occurrence of one word iiffiuences the occurrence of the next.

17.11. If there are fi samples of Wj individuals each, of n^, etc.,

Ns^==pq(^+^+ . . .4
.i-tl

684 THEORY OF STATISTICS

17.12. Standard error of expected proportion =23 *05 per cent.

Standard deviation of actual distnbution=23*09 per cent.

17.13. Standard deviation of simple sampling 23*0 per cent. The actual standard
deviation does not, therefore, seem to indicate any real variation, but only fluctuations
of sampling.

17.14. o-^szfipq as if the chance of success were p in all cases (but the mean is w/2,
not pn).

17.17. Mean number of deaths per annum — o-q® =680,

0-^=566,582' t=0*000029

CHAPTER 18

18.1. P«0'1773.

18 2. P=0-9595.

18.3. Median ' Estimated frequency= 1554 Standard error 0*28 lb.

LoiJverg : frequency 1472. Standard error 0-26 lb

Upper Q : frequency 1116. Standard error 0*34 lb.

18.4. 0-18 lb.

18 5. 0-24 lb , 14 per cent less than the s e. of the median.

18.6. Estimated frequencies * Qi=67,548, ilfi=63,152, 03=30,488.

Standard errors (years) 0*011, 0*013, 0*023.

18.7. Standard error of mean= 0*015 years.

18.8. Standard error of quartiles 0*020 years.

18.9. -4=Xl -34270.

18.10 ei2=l’36 shilhngs. Difference of means 2 shillings. Difference hardly
suggestive of real effect.

18.12. Yes, one might, because the results on farms m successive years are correlated,

18.13 Mean =5 613 ; s e. of mean 0*10.

Median =8 128; s.e. of median 0*21.

18.14. P=0*309.

18.15. ;f450,000; 350,000.

18.16. 0*12 inch.

CHAPTER 19

19.1. Standard error=0*223 Ib.

On basis of normal distribution=0* 170 lb

19.2. 0*011, 0*014.

19.3. S.e. of s.d.=0-707-7=

S.e. of 0. =0*787~7=
yn

19.4. Difference of s.d.'s 0*2. On the assumption of normality €32=0*088, Differ-
ence might therefore arise, rather infrequently, as sampling fluctuation,

19.5. f=— 0*008 for height distribution, #'=+0‘71 for marriage distribution.

ANSWERS TO THE EXERCISES

685

19 6. var Ai=—

^ n

var for normal curve

^ n n

-i 6cr« ^

var ^ = — for normal curve

n 71

■var A4=^{ + (/tj-ju,® -S/ts/tj)

+ 16 /£j/t 5 *- }

240-8

=='"'::r~ lor normal curve

19.7. For the 6th and lower moments.

19 9 Standard errors are 0-0176, 0-0158, 0-0263, and results might all have ansen
from an uncorrelated population, if the population were actually uncorrelated, the
standard errors would be the same to the number of places given, owing to the smallness
of r.

19.10. Standard errors 0-0758, 0- 1308,0 0850, and the correlations are all significant.

CHAPTER 20

20 1 x^=5-811, i/=7, P=-0-56.

20 3. v=9, P«:0-89 The hypothesis seems reasonable.

20.5. x^=27-94, v== 4, P=0* 000012. The association is significant

20.6. ;jt:®~0-7080, v=l, P=0-400. The divergences from expectation may well
have ansen by sampling fluctuations.

20 7. Use the result that for large n, is distributed approximately normally.

20 8. x*=27*68, v= 4, Ps^O-00001. The data are very suggestive of association.

20.1 1. = 13- 15, v=2, Ps=0* 0014. This is rather low and we suspect the sampling

to be non-random.

20 12. x®=9-993, V— 3, P= 0-018 Not a very good fit. (In this Exercise the
last four frequencies have been grouped together and v reduced by umt\’ to allow for
the estimation of the mean of the I'oisson distribution.)

20.14. ;i^2=0*4700, p=3, P=0-943 (by direct calculation)

20.16. If the total number of births is spread over the period evenly (on the basis
of number of days in the various months) the theoretical frequencies are 50,349, for
a month of 31 days, 48,724 for a month of 30 days and 45,476 for February.

333 • 9 and deviations cannot be due to chance.

CHAPTER 21

21.1 -0*664, r=9, Pc=0-738.

The probabihty that we should get a value of t greater in absolute value is 0-524.

21.2. The differences in the returns, including cost of manure, have mean~l,
1-375, /= 1-907, v=4, P» 0-935. Assuming that distnbution of differences

IS normal, a greater value would arise about 65 times in 1,000 There is some reason
for supposing that the increased returns on the better manured plot are real, and that
it would therefore pay to continue the more expensive dressing.

21.3. Applying the / test for two samples,

^«0-0991. vr=i4, P=:;=0*54

There is nothing in this test to suggest that populations were unlike as regards height,

686

THEORY OF STATISTICS

21 4. 1761, 1 'l =*9. 1/^=5. The difference of standard deviations is not significant.

Coupled with Exercise 21.3, we conclude that there is no ground for supposing the
two populations different as regards height.

21.5. Applying the / test for two samples,

2!=2-683. v=:4, P«a-972

The difference of means is hkely to be significant, which supports the suggestion.

21 6. logo = o‘=;^^^2~0*2887

The observed deviation is suggestive, but not decisive.

21.8, P—0-0048. For the standard error formula P=0* 0000078.

21 9. All significant.

CHAPTER 22

The analysis is

Sum of squares

d,f.

Quotient

Between batches .

44,360

14,787

Residual

. 151,351

6,880

Total

. 195,711

7,828

» 5= 0*383 which

IS not significant.

The analysis is

Sum of squares

d.f.

Quotient

Between consignments . 9-71

1*94

Between observers

13*13

4*38

Residual

13*12

0*87

Total

35*96

Differences between observers are significant at the 5% level,

22.5. All significant.

22.6. Significantly non-linear.

22.8. The analysis is

Sum of squares

d.f.

Between investigators

775

Between areas ,

239

Residual .

1,175

Total

2,190

Differences are not significant.

CHAPTER 23

23.6. (a) 0-0726, (^>) 0-0553, (c) 0-0661, (d) 0-0482,

CHAPTER 24

24.1, 0*93877, 0-93823, 0-93822.

24.2. 0*823632, 0-818050, 0-817939. The inclusion of the third difference affects
only the fourth place by a single unit, so we can probably trust ihe answer to four figures.

ANSWERS TO THE EXERCISES

687

24.3. Usmg logarithmic iaterpolation, the successive approximations are: 0*11200,
0*10044, 0*09963. Second diference interpolation usmg the last three data only
gives 0*09859. It looks as if we could trust the figure as about 0*100 or 0*099.

24.4 4195, 4443, 4724, 5036, 5380.

24.7. 11*388 approximately.

24.8 Median 4*8924, 4*8869 First decile 1*9474, 1*9572. Ninth decile 8*4286,
8*3733. As we would probably state such figures only to two decimal places, the
median would not be appreciably afiected by taking second differences into account,
but the deciles would be slightly corrected.

24.9. Maximum at 1*336, or day 40, 25th July, value 63*7.

Minimum at 1*184, or day 35*5, 20th-21st January, value 38*0.

These estimates are very poor. The maximum is actually 63*4 on 15th- 17th July,
and the minimum 37*9 on 8th- 12th January.

CHAPTER 25

25.1. Index numbers are

1923 100

1927

100

1931

4 101

5 101

6 100

1930

25,3. Index numbers are

(1)

(2)

1930

100

102

103

25.4. To nearest unit, index is

102 in all cases.

25.6. Index numbers are

(1)

(2)

1935

100

101

100

109

110

103

105

106

107

1940

134

131

134

131

141

138

146

144

CHAPTER 26

26.1. The figures are given in Table 27,1, page 640.

26.3, To the nearest unit the first average gives —

73 (1924), 72, 71, 71, 68, 67, 66, 66, 63, 62, 61, 59, 57, 56, 56, 56, 56, 56,
54, 52, 49 (1944).

A second average of these figures gives-^

71 (1926), 70, 69, 68, 66, 65, 64, 62, 60, 59. 58. 57, 56, 56. 55, 55. 53 (1942).

688

THEORY OF STATISTICS

26 4 Expressed as a percentage of the average monthly rainfall the figures are —

1943

1944

1945

1946

Jan.

, 207

114

Feb.

121

132

Mar.

‘ 56

April

104

May

. 139

139

130

June

102

131

139

July

108

Aug.

161

Sept.

. 134

165

193

Oct.

113

103

Nov.

175

178

Dec.

105

102

4.0*735,

+ 0-367, .•j=+0-054, r.= -0

•102,

-0*082,

•4-0*027.

26.7. The weights of the process are

i [1, 3, 6, 9, 12, 13, ... ]

26.8. The index-numbers are —

1926

126

1936

108

102

119

103

26.9. As a prehminary show that for a cubic curve (third differences constant)

^ [k] ut ^ ut {ut^i — 2ut 4 - ut^\)

26.10. The index-numbers are —

Quarter

12 3 4

1928

120

118

117

115

112

109

104

26.12 See the hint on Exercise 26.9.

CHAPTER 27

27.1. The number of turning pomts is 31, almost exactly the expected number 30.67.

27.2, When /tK=0, the mean-distance is 3, the known result for random senes.

27 3 The mean distances aie 7 28, 4*96 and 4*96 and the autoregressne peruxls
are 10-90, 7*24 and 5*68, respectuely

ANSWERS TO THE EXERCISES

689

27.5. a=-l-206, 6 = +0-420.

27.10. The autocorrelations

are as

follows —

0*957

~0'053

0*836

-0*030

0*660

-0*012

0*461

-0 002

0*269

0 003

0*111

0*003

0*000

0*002

-0*061

0*001

-0*082

0*000

-0*074

0*000

INDEX

[The references are to pages. References to Greek letters follow those for Roman letters]

Absolute measures of dispersion, 144

Accident, death from, 193

Achenwall, G , footnote, xvii

Additive propcrtv of yf, 473

Ages, at dcacli iiom scarlet fever (Table ^

4.11) , 89; (Fig 4 11), 90

— , at death from all causes (Table 4.16),

97 , (Fig. 4.17), 97

— , of cows correlated with milk-yield, see
milk-yield

— , of husband and wife (Table 9.2), 201 ;
constants, 229 ; correlation ratios
(Exercise 11.2), 279

Agricultural labourers’ earnings, see
Earnings

, minimum wage-rates, 128 , means

and s. d , 128-130 , median and m d ,
139 ; quartiles, 141

Agricultural Market Report, data cited
from, (Table 9 7), 207
Agricultural Price Index, (Example 25.3),
603

Agricultural Statistics, data from, (Table
13.1), 311; (Table 23.1), 545
Ammon, O., data cited from, (Table 3.2),

ApdTysis of variance, 503-529 ; for a
single classification, 503 ;
relationship with mter-class correlation,
512 ; for two-fold classification, 513 ;
significance of correlation ratio, 517 , of
linearity of regression, 519 ; of multiple
correlation, 521 ; unequal numbers m
classes, 512 ; three-fold classification,
523 , of family budgets (Example 23 7),
546

Animal feeding stuffs, index numbers of
prices of (Table 9.7), 207 ; (Figure 9.4),
211 ; correlation, 223-5
Annual values of estates in 1715, (Table

4.12) , 94 ; (Fig. 4.13), 92
Arithmetic mean, see Mean, anthmetic
Array, definition, 199 ; type of, 199 ; s.d.

of, 221 , homo- and hetero- scedasticity
of, footnote 221 ; m normal correlation,
>237. 241, 305

Cssociation, generally, 19-48 ; definition,

22 ; testing for, 24-28 ; coefficient
of, 30 ; partial, 31-37 ; illusory, 37-8 ; in
incomplete data, 38-40 ; complete
independence, 40-1

691

Asymmetrical frequency-distributions 83-
90 ; relative position of mean, median
and mode m, 117 See also Skewness
AtJ^rcrStion, in correlation, 313
Attributes, generally, 1-18 , class-fre-
quencies 3-6 ; positive, 7-9 ; consist-
ence, 9-11 , incomplete data, 11-14
Australian marriages, distribution of
(Table 4 8) 84 ; (Fig 4.8) 85 , mean
and s d. 132 , third and fourth
moments 157 , and 160 ; median
and quartiles 163 , skewness 163 ;
kurtosis 164 , standaid error of mean,
median and quartiles (Exercises 18 6
and 18 7) 435 , standard error of s. d.,
444 , correlation between errors in
mean and s. d (Exercise 19 5), 457
Auto-correlation, see Serial correlation
Auto regressive senes, 645-658 , estima-
tion of constants, 649 , properties of,
655-8. See also Correlogram, Serial
correlation

Averages, generally, 102-124 ; desirable
properties of, 103-4 , forms of, 104 ;
average m sense of arithmetic mean,
105 ; See also Mean, Median, Mode
Axes, prmcipal, in correlation, 242, 323,
362-3

Babington Smith, B , factor analysis,
323 ; random sampling numbers, 376.
Barlow’s Tables of Squares etc , 56
Barometer heights, (Table 4.10) 88; (Fig.
4-10) 88 ; means, medians and modes of,
117 ; modes of, 583
Barley, prices of, (Table 25 2) 593
Base-year, in index numbers, 591-2
Bateson, W., data cited from, 29
Beetles {Chry$omeUd<s) , sizes of genera
(Table 4.13), 95

Bernoulli, James, Binomial distribution,
169

Bertrand, J.L.F., Quotation on chance,
374

“ Best fit,” see Least Squares

Beta- function, 494

Beveridge, Lord, 645

Bias, in samplmg, 371-4, 375-6, 531-2,

— m estimation. 544-547, 550-3 ; tech-

692

THEORY OF STATISTICS

meal definition, 547-9 , cumulative
effect of, 549
— , in scale reading, 74
Biehl, K., data cited from, 315
Bielfeld, Baron, J. F. von, use of word
statistics ”, XVI

Binomial distribution, 169-195 , genesis
of, 169-171 , form of, 172-174 , con-
tents of, 174-6 , mechanical representa-
tion of, 176 , limiting form, 177-181 ,
Poisson distribution, 189-191 ; in
sampling of attributes', 386-394, see
Sampling of Attributes
Birth-rate, in local government areas, 70 .
correlation with number of births, 206,
constants of distribution (Exercise 9 3)
234 , standardisation of, 333-7
of cattle, (Table 26 3), 614
Bivariate distribution, 201 , normal sur-
face, 237-250 , see also Correlation
Blackman, V H , data on duckweed, 350
Bortkiewicz, L von, Poisson distribution,
193

Breaking-up a group, in interpolation,
571-3

British Association, data cited from,
(stature. Table 4 7) 82 , (weight. Table
to Exercise 4 6), 100

Cambridgeshire, mortality in, 561
Cards, punched, for recording of data, 62 ,
for sampling, 375

CaxToll, Lewis (pseudonym), (Exercise 1 9)
16

Cells, in 459

Census data, see Registrar- General
Centred averages, 624
Chance, see Randomness, Probability
Charlier, C. V L , in sampling theory, 407
Chi-square, chi-squared, see
Cholera and inoculation, 25, 27, 467, 473
Chrysomehdae, see Beetles
Circular test, m index-numbers. 602
Clark, R D , data from, 194
Class, in theory of attributes, 2-4 ; class-
frequency, 3 , ultimate classes, 5-7 ,
positive and negative classes, 3
Class-interval, definition, 70 ; choice of
• magnitude and position, 72-4 ; see also
Sheppard’s Corrections
Classification, generally, 1-2; by dicho-
tomy, 2-3 , manifold, 49 , horn osfen eons
59-61 : as senes of di-hotorruLa 61 , ])\
punched cards, 62
Closeness of fit, see

Cloudiness at Greenwich, (Fig 4.15), 93,
(fable 4,14) 96,

Coefficients of association etc, see under
Association etc.

Complex frequency-distributions, 92
Confluence analysis, 323

Consistence, of class-frequencies, 9-11 , of
correlation coefficients, 301-2
Constraints, in Lexis’ sense, 408 , in y^,

461

Contingency, coefficient of (Pearson’s) 53,
(Tschuprow’s) 54 , isotropy in, 57-9 ;
relation with normal correlation, 250 ,
standard error of, 454
— , tables, definition, 50 , association in,
51-2 ; isotropy in, 57-9, 248 , indepen-
dence in, 59 , degrees of freedom in,

462 , tests of divergence from indepen-
dence, 467-8

Corrections, for grouping, see Sheppard’s
Corrections

— , of correlations for errors of observa-
tion, 328 , of death-rates, 335-7
Correlation, generally, 199-339 , con-
struction of tables, 199-201 , representa-
tion of tables by diagrammatic methods,
203-212 ; treatment as contingency,
212 , for illustrations see Frequency-
distributions, Illustrations
Product-moment coefficient, defini-
tion, 218 , lines of regression 214-8 ,
calculation of, 222-230 , corrections for
grouping, 231 , estimation of, 253-5 ,
modifiable umt, 310-2 ; attenuation of,
313-5 , nonsense correlations, 315-7 ;
errors of observation in, 328 , between
indices, 330-1 , heterogeneity of
material, 331 , standard error of, 451-2;
” 'u samples, 495-9

Uc.n- - » - J. ... 258-268, see Rank-

correlation , grade-correlation, 268-9 ,
tetrachoric correlation, 270-1 , intra-
class correlation, 272-7
— , normal, 237-252 , linearity of regres-
sion in, 240-1 , homoscedasticity in,
241, isotropy in, 248-250, relation
wuth contingency, 250 , multivariate,
303-6.

— , partial, 281-306, generalised regres-
sions, 282 , notation, 284 , expression
in terms of lower order coefficients, 290 ;
calculation, of, 290-7 ; expression in
terms of higher order coefficients, 300-1;
fallacies in interpretation, 302-3 ; test
of significance, 451, 495-9
— , multiple. 281, 361 , coefficient of, 298-
300 ; significance of, 453, 521-2
— , ratios, 256-8 ; relation with goodness
of fit, 361 , significance of, 453, 517-9
— , serial, in time-series, 639
Correlogram, definition, 651 ; of auto-
regressive and harmonic senes, 651-4
Cosm, value of estates m 1715 (Table 4.12),
94

Cost of living index, 596
— , of electncitv, see Electricity
Co\ariance, definition, 222
Coutts, J R H , data cited from (Table
15 5), 356.

INDEX

693

Cows, distribution according to milk-
yield, see Milk-yield

Criminals, weights and mentality, (Table
3 6), 64

Crop forecasting, pessismism in, 544-5

Crops and weather, correlation of,
320-1

Cumulants, definition, 164-5

Cumulative frequency (distribution)
function, 144-6

Curve fitting, generally, 340-363 ; least
squares in, 342-3 ; equations for, 344 ,
calculation, 346-8 ; reduction to linear
form, 348 , residuals, 360 ; closeness of
fit, 361

Curvilinear regression, see Regression

Darbishire, a D., data cited from, 121 ,
(Exercise 17 12), 411

Datura, association in, 29 , (Exercise
20 6), 479

Davenport, C B , data cited from, (Table
9 1), 200

David, census of Israelites, footnote, xiv
David, F N., on correlation coefficient,
495, 496

Deaf-mutism, association with imbecility,
(Exercises 2.1 and 2,15), 43, 46 , fre-
quency among offspring of deaf-mutes,
(Exercise 4.5 (&)), 99

Deaths or death-rates, association with
occupation, 39 ; from scarlet fever
(Table 4.11, Figure 4 11), 89, 90;
infantile and general mortality, 317-9 ,
standardisation of, 39, 335-7 ; from
accidents, 396 ; from explosions in
mines 406 ; m non-simple sampling,
396, 404, 406 ; mortality in Cambridge-
shire, 561

Deciles, definition, 144 , standard error of,
423

Defects in schoolchildren, 5-6, 33-5
Degrees of freedom, in 461-2 ; in
analysis of vanance, 484-5, 505 ; in
#-test, 487.

Demoivre, A , discoverer of normal distri-
bution, 169

Dependent variable, in regression and
curve fitting, 282, 345
Design of statistical inquiries, 370, 530-
551 ; see Sampling

Deviance, definition, 504. See Analysis
of vanance

Deviation, mean, 137-140 ; corrections for
groupmg (Exercise 6.11) 149, le^t
about median, 138 ; comparison with
standard deviation, 140 ; of normal
distribution, 184
— , quartile, see Quartile
— , root-mean-square, see Deviation,
Standard

Deviation, Standard , definition, 126 , re-
lation with root-mean-square deviation,
127-8; calculation of, 128-133 , correc-
tion for groupmg, 133-4 , prop^rt^es of,
134-137 ; of senes of natunil ri ambers,
136 , of rectangular distribution, 136,
of arrays in correlation, 221, 240-1,
243 , generalised, 284 , of sum or
difference, 326-7 , mfiuence of errors of
observation on, 327 , of an index, 329 ,
of binomial distribution, 175 , of
Poisson distribution, 191. See also
Error, Standard

Dice, records of throws, (Table 4 15 and
Figure 4.16), 96; (Exercise 8 2), 197;
divergence from expectation, 387-8, 389,
(Exercise 17 1), 409, 466, 470-1, 474-6.

Difference-method, in correlation and
time-senes, see Vanate-difference

Differences, in interpolation, 556 and see
Interpolation

Discounts and reserves in American banks
(Table 9 5, Figure 9.2), 205, 209

Discriminant functions, 606

Dispersion, measures of, generally, 125-
150 , absolute measures of, 143 ; m
Lexis’ sense, 407-8 , see Deviation,
Mean ; Deviation, Standard , Range ;
Quartiles

Distance-velocity relation in nebulae,
340-1, 347-8, 362

Distnbution curve, 144-6 ; of frequency,
see Frequency-distribution.

Duckweed, correlation in, 227-230, growth
of, 348-350

Durant, H. E., on sampling for reading,
550

Earnings of agricultural labourers, corre-
lation m, (Exercise 9 2) 233 , partial
correlation, 290-3, (Figure 12 1) 297
Economy in vanables, 322
Edgeworth, F. Y., data on dice-throwmg,
(Table 4,15), 96
Efficient estimates, 475
Egg-pnces, index-numbers of, 625
Electoral voting in English municipalities,
(Table 17.1) 402

Electricity Commission, data from returns
of, (Table 15.4), 353

Electricity, costs and numbers of units of,
(Table 15.4), 353, 350-2
Elimination of seasonal effects in time-
senes, 624-5

Engledow, Sir Frank L., data from, (Table
22.5), 509

Error function, see Normal Distribution.
Error, mean, 137
Error, mean-square, 137
Error, probable, 137, 390 ; see Error,
Standard

694

THEORY OF STATISTICS

Error, Standard, definition 390, 421 , ot
number or proportion of successes, 387 ,
when sample-numbers vary (Exercise
17.11), 411 , when chance of success is
small, 393, of percentiles, quartiles
etc , 42v3 , of semi-interquaitile range,
427 , of arithmetic mean, 428 ; of
variance, 442 ; of standard deviation,
442 ; of coefficient of variation, 448 ; of
moments about a fixed point, 437-9 ; of
moments about the mean, 440 , of third
and fourth moments about the mean,
447 , of Pi and *^50 , of coefficients
of correlation and regression, 451-3 ,
approximate formulae for correlation
ratio and multiple correlation, 453 , of
coefficient of association, 454 , of

mean-square contingency, 454 , of

Spearman’s p, 454 , of Kendall’s r, 455.
See also Sampling, Theory of
Error, Theory of, see Sampling, Theory of
Estates, value of, see Value
Estimates, precision of, 369 , efficient,
475 , m small samples, 482 ; of anth-
metic mean, 482-3 ; of variance, 484 ,
degree of freedom of, 484-5
Estimation, Theory of, 369 , of theoretical
frequencies in ^®st, 474-5 , of
position of maximum, 582 ; of con-
stants m autoregressive senes, 649
Exammation of samples, 530, 544-552
Existent populations, 367
Explosions in coal-mines, deaths from, 406
Eye-colour, association of, father and son,
26-7, (Exercise 2 4), 44 (Table 3 4), 58-9
of husband and wife (Exercise 2 5), 44 ;
with hair-colour (Table 3 2), 50

FACTOR-analysis, 323

Factor reversal test, in time-series, 601

Fallacies, m interpreting associations, 37-8,
due to change in classification, 60-1 , in
interpreting correlations, 302-3 , spun-
ous correlation, 330-1 ; due to hetero-
geneity, 331-2 , nonsense-correlations,
315

Family budget data in Nagpur, 546-7

Farm survey, estimation by sampling
fraction, 537-8

Fay, E. A., data from, (Exercise 4,5 (5) ),
99

Fecundity of brood-mares, (Table 4.9,
Figure 4 9), 86, 87

Finite populations, 367 , variance of
proportion from, 405

Fisher, Irving, 601

Fisher, R. A., Tables of 465 ; hmiting
normality of 469 ; Tables of f, 488-
493 ; distnbution of vanance-ratio, see
Fisher’s distribution; distribution of

correlation coefficient, 495 ; trans-
formation, 497

Fisher’s distribution (^-distribution) 493 ;
in analysis of variance, 506 , for large
numbers of degrees of freedom, 512 , see
also Analysis of Variance
Fit, goodness of,

Fitting, of curvesi see Curve fitting
Flying bombs, distribution of, 194
Food. Drink and Tobacco trades, sizes of
firms m, (Exercise 4 5 (a) ), 99
“ Footrule,” Spearman’s, footnote, 262
Forecasting of ''ro 'ds,

Fourier analysi.^ ' - I [ . ‘ . analysis

France, Anatole, xiv

Freedom, degrees of, see Degrees of
Freedom

Frequency-curve, 80-1 ; ideal forms of, 81,
84, 91, 93 ; Pearson’s, 194-5
Frequency-distributions, generally, 69-
101 , magnitude and position of class-
intervals, 72-4 ; graphical representa-
tion, 78-81 ; common t 5 q)es of, 81-92;
symmetrical, 82 ; skew, S3-7 ; J-
shaped, 87-90 , U-shaped, 90-1 ,
truncated forms, 91-2 , complex forms,
92-4 , pseudo-forms 94-8 , i eduction to
absolute scale, 144 ; cumulated sum
(distribution curve) 144-146 ; theo-
retical forms, 169-198. See Normal
distribution, Binomial distribution,
Poisson distribution, Correlation,
Bivariate, Multivariate distribution
Frequency-distnbutions, illustrations :
birth-rates m local government areas
(Table 4.1), 70 , capsules of poppies
(Table 4.2), 71 , lengths of screws
(Table 4 3), 72 ; final digits in measure-
ments (Table 4.4), 74 , persons liable
to surtax and super-tax (Table 4 5), 77 ;
headbreadths of students (Table 4.6),
78 ; statures of men (Table 4 7), 82 ,
marriages in Australia (Table 4.8), 84 ;
fecundity in brood mares (Table 4 9),
86 ; barometric heights (Table 4 10), 88,
deaths from scarlet fever (Table 4 11),
89 ; values of estates (Table 4.12), 94 ;
beetles (Table 4 13), 95 ; cloudiness
(Table 4 14), 96 ; dice- throws (Table
4.15), 96; male deaths (Table 4 16),
97 , size of firms in Food,
Drink and Tobacco Trades (Exercise
4 5 (a) ), 99 , deaf-mutes (Exercise 4.5
(b ) ), 99; yield of gram (Exercise 4 5
(l) ), 99 ; petals m buttercups (Exercise
4 5 {d) ), 100 , weights of men ^ Exercise
4.6), 100 , deaths from horse-kick, 193 ,
flying bombs, 194

Diameters in shell fish (Table 9.1),
200 , age of husband and wife (Table
9 2), 201 ; statures of father and sou
(Table 9.3), 202 , age and milk-yield of
cows (Table 9.4), 204 ; discount ratio

INDEX

695

and reserves in banks (Table 9 5), 205 ,
birth-rate and numbers of births (Table
9.6), 206 ; fronds of ferns (Table 9.9),
226 , electorate voting in muncipalities
(Table 17 1). 402.

Frequency-polygons, 78-80
Frequency-surface, see Bivariate Distribu-
tions

Fnsch, R., confluence analysis, 323
Fundamental sets, specifying data, 6

Galton, Sir Francis, ogive, 145 ; bino-
mial apparatus, 176 ; regression, 213 ;
data cited from, 26, (Exercises 2.4 and
2 5), 44, (Table 3.4), 58
Gauss, K F , normal distribution, 169 ;

use of term “ mean error,” 137
Gehlke, C. E , data cited from, 315
Geometric mean, see Mean, geometric
Gini, C , coefficient of mean-difference,
146-7

Goodness of fit, see
Gosset, W. S., see ” Student.”

Grades, 144 ; grade correlation, 268-270 ,
see Quartiles

Graduation, 575-9. See Interpolation
Graphical methods, of representing fre-
quency-distribution, 78-80 ; of mter-
polating for quartiles, 113-4 ; of rep-
resenting correlation (scatter diagram),
211-2 ; of estimatmg correlation, 253-4,
Gray, J , data cited from, 398
Greenwood, M , data cited from, 25-6,
(Table 8 3), 175

Group, breakmg-up of, in interpolation,
571-5

Groupmg-corrections see Sheppard's
corrections

Grouping of observations, in frequency-
distributions, 71-5 ; in correlations,
310-3

Growth of duckweed, 348-350

Hair and eye colour, contingency, (Table
3.2), 50, (Table 3 3), 55 ; non-isotropy
of, 56-7 ; in school-girls, 398, 400
Half-invariants, see Cumulants
Hall, Sir Daniel, data cited from (Exercise
4.5 {c ) ), 99

Halving a group, m interpolation, 573-5
Hannan, H., Factor Analysis, 323
Haimonic analysis, 641-5
Harmomc mean, see Mean, Harmonic
Head-breadths of students, (Table 4.1
Figures 4.1, and 4.2), 78.9
Height, of men, see Stature
— of wheat plants, (Table 16-1, Figure
16.1), 372

Heteroscedasticity, footnote, 221

Histogram, 78-80

Hollis, T., data cited from, (Table 4.12),
94

Holzmger, K J , Factor Analysis, 323
Homoscedasticity, footnote, 221
Hooker, R. H , investigation into weather
and crops, 320-1, (Exercise 12.1), 308
Houghton, C. T , data cited from, 603,
625-7

Houses, inhabited and uninhabited,
(Exercise 3.2), 66

Hubble, E , data cited from, (Table 15.1),
340

Human bias, in sampling, 371-4 530-2,

542-551

Humason, M. L , data cited from, (Table

15.1) , 340

Husbands and wives, correlation between
ages (Table 9.2), 201 ; constants of,
229-230 , correlation ratios, (Exercise

11.2) , 279

Hypothetical population, 367 ; sampling
from, 380-1

Ideal index-number, 601
Illusory associations, 37-8
Incomes, see Surtax

Independence, of attributes, 19-22 ; com-
plete, 40-41 ; m contingency tables, 51,
59 ; test for, 467-9, 472 ; of variables,
213

Independent variable, m curve fitting,
345-6

Index-numbers, generally, 590-609 ; price
mdex-numbers, 592-4 ; geometric
means, 595-9 ; time-reversal test, 599-
601 ; factor-reversal test, 601 : “ideal”
number, 601 , circular test, 602 ;
Imkmg methods, 603-4 ,* quantum
mdices, 605-6 ; of animal feeding stufis
and oats, (Table 9.7, Figure 9.4), 207,
211, correlation, 223-5 ; of egg-pnces,
625-7 ; of wheat-prices, 645 (Figure
27 1), 644

Indices, correlation between, 330
Infinite populations, in samphng, 367, 380
Inoculation against cholera, see Cholera
— , against tuberculosis in cattle, 472
Intensity, in periodogram, 642
Interactions, in vanance analysis, 524
Intenm index of retail prices, 596-8
Interclass correlation, 273
Interpolation and graduation, generally,
55^589 , differences, 556-8 ; Newton’s
formula, 558-561 ; of statistical series,
561-7 ; effect of errors on differences,
567-571 ; subdividmg an interval, 571 ;
brealdng up a group, 571-5 ; graduation,
575-9 ; inverse, 579-582 ; estimation of
a maximum, 582-3

6g6

THEORY OF STATISTICS

Interval, subdivision of, 571
Intraclass correlation, 272-7 , relation
with variance-analysis, 512-3.

Inverse interpolation, 579-582
Isotropy, definition, 57 , generally, 56-9 ,
of normal distribution, 248
Isserlis, L , on index-numbers, 603

J-SHAPED frequency-distributions, 87-90
Jute, sampling of, 531, 545-6
Juvenile delmquency, 315

Kelley, T. L., Statistical Tables, 297
Kelvin, Lord, dictum on measurement, xiii
Kendall, M G , on factor analysis, 323 ;
random sampling numbers, 376 , rank
cgrrelation, 455, 456 , data from,
(Table 27 2), 647, (Table 27.3), 648,
peaks in time-senes, 657
Kick of a horse, deaths from, 193
King, G , graduation of age statistics, 577
Kurtosis, definition, 164 ; of binomial,
175-6 ; of normal, 183 , of Poisson, 192 ;
effect on standard error of standard
deviation, 443-4

Labourers, agricultural, see Agricultural,
Eammgs

Lanarkshire milk experiment, 543
Laplace, P. S , ]\Iarquis de, normal distri-
bution, 169

Leading term and leading differences, 557
Least squares, method of, m regression,
216-7, 282-4 ; m curve fitting, 343-5
Lee, Alice, data cited from, (Table 4 9),
86; (Table 9 3). 202

Lemna minor, correlation in, (Table 9 9)
226 ,* growth in, 348-350
Leptokurtosis, 164

Levels of s’gn.ficance, in ^est, 471-2 ;

in f-test, 488 ; in ^--test 510-2
Lexis, W , use of term “ dispersion,*’
407-8

Linear constraints, 451
Linearity of regression, see Regression
Linking methods in index-numbers, 603
I.ittle, W , data cited from (Exercise 9.2),
233

Lloyd's Register, data cited from, (Table
26 2), 613

Loss in weight of soils, see Percentage
Losses of ships, (Table 26 2, Figure 26.2),
613-4, 639

Lottery sampling, 375

Macdonell, W. R., data cited from (Table
4 6), 78

Mahalanobis, P C , data cited from, 529,
531, 546, 549

Manunal treatments, 515, 524
Marley, Joan, data cited from, (Table
26.3), 614

Marriages, Australian, see Australian
— , age at, (Table 22 2), 507
Maximum, estimation of position of, 582
Mean, arithmetic, generally, 104-111 ;
calculation of, 105-8 ; properties of,
110-1 ; relation with mode and median,
117; of sum on difference, 110-1;
reciprocal relation with harmonic mean
121 ; of binomial, 174 , of Poisson, 191 ;
weighting of, 332-7 ; standard error of,
428 , means of two samples, 429-430 ;
/-test for, 487-492 ; estimates of, 482-3
Mean deviation, see Deviation, mean
— , difference, 146-7
— , error, 137

— , geometric, 118-120; weighting of,
337 ; in mdex-numbers, 595-6, 598-9
— , harmomc, 120-1 ; relation with
arithmetic mean, 121 , m sampling
theory (Exercise 17 11), 411

— square contingency, see Contingency

— square error, 137

— , weighted, 332-7 ; in death-rates etc.,
335-6

Median, generally, 111-116; determina-
tion of, 112-4 ; comparison with mean,
114-5; advantages of, 115-6; relation
with mean and mode, 117 ; standard
error of, 421-6

Mendehan breeding experiments, 29, 121,
389

Mental defectives, relation with radio
licences, (Table 13.2), 315
Mentality, relation with weight in
criminals (Table 3 6), 64
Mercer, W. B., data cited from (Exercise
4 5 (c) ), 99

Method of least-squares, see Least squares
Mice, numbers in litters, 121, (Exercises
17.12 and 17.13), 411

Milk-yield in cows, correlation with age
(Table 9 4), 204 , (Figuie 9 9), 219 ;
constants of (Exercise 9.3), 235 ;

correlation ratios (Exercise 11 1),) 278
Milton, John, use of word " statist ”, xvi
Mode, generally, 116-7 , relation with
mean and median, 117; estimation of,
582-3

Modifiable unit, 310-3
Modifying central ordinates, 583-4
Modulus, as measure of dispersion, 137
Moments, first, definition, 106 , second,
definition, 127 ; generally, 151-160 ;
about mean m terras of those about any
oint, 152-3 , calculation of, 153-8 ;
heppard corrections for, 158-9 , of

INDEX

697

bivariate distributions, footnote, 222 ;
standard errors of, 437-442 , correlation
between errors 441-2

Moore, L. B., data cited from, (Table 4 9),
86

Mortality, see Death-rates
Moving averages, 617-624 ; see Trend
— , weights, in index-numbers, 603
Municipal elections, (Table 17 1), 402
Multiple ’correlation, see Correlation,
multiple

Miiltnanate anahsis, 323

National Income, (Table 26.7), 628
Newbold, Ethel M , partial correlations,
306

Newton’s formula, in interpolation, 558-
561 , bmomial coefficient in (Table
24.4), 564

Nonsense correlations, 315-7
Normal dispersion, in Lexis" sense, 407
Normal distribution, as limit of binomial,
177-181, properties of, 181-3, constants
of, 183-4 , ordinates and areas of,
184-6 ; as an error distribution, 186-7 ;
occurrence of, in nature and theory,
187-9 ; normality of sampling distri-
butions, 485-6

Norton, J. P., data cited from (Table 9.5),
205

Oats, correlation of prices with those of
home-grown feeding stuffs, (Table 9,7)
207, (Figure 9.4) 211, 223-5; price
index-numbers of, (Table 25.2) 593
Ogbum, W F , data cited from, 322
Ogive, Galton’s, 145 ; see Distribution
Curve

Order statistics, 260. See Rank Correla-
tion

Oscillations in time-series, 614-624 ;
effect of moving averages on, 629-631 ,
generally, 637-658 ; serial correlation,
639-641 ; periodogram analysis, 641-5 ;
autoregressive senes, 645-651 ; correlo-
gram, 651-654 ; periods ” of, 656-8
Orthogonal polynomials, 357
Oscuiatory interpolation, 579

Parabolas, fitting of, 341 ; see Curve
fitting

Parameters, definition, 414
Partial association, see Association, partial
— , correlation, see Correlation, partial
— , rank correlation, 264, 306

Pauperism, correlation of, (Exercise 9.2),
233 ; 290-3, 297

Peak, in time-senes, 638 , mean-distance
in autoregressive senes, 656-7
Pearce, Gertrude E., data cited from
(Table 4 14), 96

Pearson, Karl, contingency, 53 ; correction
to coefficient of contingency, 54 ;
definition of fi's, footnote, 164 , bino-
mial apparatus, 176 , system of
curves, 94-5 ; normal correlation and
contingency, 250 ; data cited from,
(Table 3.4), 58, (Exercise 3.1), 65,
(Table 4 9), 86, 117, (Table 9 3), 202
Pearson curves, 94-5
Peas, experiments in crossing; 389
Pecien, correlation between two diameters
of shell, (Table 9.1), 200 ; constants of,
(Exercise 9 3), 235

Percentage loss of weight in soils, (Table
15.3), 356 ; curve fitted to, 352-6
— , standard error of, 387
Percentiles, see Quantiles
Period, of time-series, see Oscillations
Periodogram, 641-5 ; see Wheat-pnces
Pessimism, in crop forecasting, 544-5
Petals, of buttercup, (Exercise 4 5 (d) ),
100 , unsuitability of median for, 112
Phase, m time-series, 638
Platykurtosis, 164

Poisson distribution, 189-194 ; constants
of, 191-2 ; in samphng, 393
Polynomials, in curve-fitting, 341-4 ;
orthogonal, 357 ; differences of, 556 ;
forms of, in interpolation, 566
Poppies, sti^atic, rays of, (Table 4.2), 71;

unsuitability of median for, 112
Population, statistical, footnote, 1
— , estimation of, between censuses, 120;

curve fitted to, 358
Positive classes and attributes, 7-9
Potatoes, yields of, (Table 13 1), 311 ;
515-6; (Table 23 1), 545, (Exercise
27.1), 659

Precision, 137 ; of estimates, 369 ; varies
as square-root of sample number, 394
Prest, A. R., data cited from, (Table 26.7),
628

Pretonus, S J., data cited from, (Table
4 8), 84, (Table 4.10), 88
Price-level, effect of change in, 627
Price-relatives, 591

Pnces, mdex-numbers of, 592-606, see
Index-numbers ; use of geometric mean
in, 120

Principal axes, in correlation, (Figure
10 1), 240 ; in curve fitting* 361
Probability, 369 ; 415-7. See Sampling
Probable error, see Error, standard
Pseudo frequency-distribution, 94-5
Punched cards, recording of information
on, 62-3

Purposive sampling, 369* 382-4

698

THEORY OF STATISTICS

Quality control, use of range in, 125
Quantiles, 144 ; standard error, 421-8
Quantum index-numbers, 605-6
Quartile deviation, see Quartiles
Quarfciles, definition, 140 , deviation, 142,
empirical relation with standard devia-
tion, 142-3 , graphical determination of,
144-6 ; in measuring skewness, 160-1 ,
of normal distribution, 186 , standard
errors of, 421-3

Quetelet, L. A. J , data cited from
(Exercise 17.2), 409
Quota sampling, 542

Random element, in time-series, 614 ,
effect of trend-elimmation on, 630-1 ,
645-6

Random sampling, 374-384 ; technique of,
374-6 ; random sampling numbers,
376-9; importance of, 381-2
Randomness, tests for, in time-senes, 638-
641

Range, as measure of dispersion, 125
Rank correlation, 260-270 , Spearman’s p,
261-2 ; Kendall’s r, 262-4 , tied ranks,
264-6 , relationship with product-
moment correlation, 269-270 , partial
correlation, 306 , standard error of p,
454, of r, 455-6 ; i-test of, 454, 493
Ranunculus bulbosus, see Petals
Registrar-General, standardisation of
death-rates, 336 ; data cited from
reports of ■ death-rates of occupied
males, 39 ; blindness and derangement
(Exercises 2 6 and 2.15), 44, 46,

cancer (Exercise 2 16), 47 ; housing
(Table 3.5) 63 ,* birth-rates, (Table 4.1),
70 ; deaths from scarlet fever (Table
4.11), 89; ages of husband and wife,
(Table 9 2), 201 ; birth-rates (Table
9.6), 206 ; general and infantile mortal-
ity (Figure 131), 318 ; population
(Table 15.6), 359 ; voting in mumcipal
elections (Table 17.1), 402 ; expectation
of life, 561

Regression, generally, 213-231 ; curves of,
213 ; coefficients of, 221 ; calculation
of, 222-230 ; in normal variation, 241 ;
non-linear, 213, 255-6 , multiple vana-
tion, 281-306; partial regressions,
281-5 ; in terms of higher-order
coefficients 300 ; in terms of lower-
order coefficents, 289 ; in wheat-yields
and weather, 320-2 ; economy m
number of variables, 322-3 ; standard
error of, 453 ; significance m small
samples, 492-3 ; test of linearity of,
519-520

Reiersol, O. on confluence analysis, 323

Reserves and discounts in American banks,
(Table 9.5, Figure 9 2). 205, 209
Residuals, 343, see Least Squares
Rider, P R data cited from, 414
Room space, deficiency in, (Table 3.5), 63

Sampling fractions, 533-9
Sampling numbers, see Random samphng
cmerally, 366-534, types of
367-8 , tests of significance,
370 ; types of sampling 370-1 , random,
371-3 ; bias in, 373-4 , technique of,
random, 374-6 , random sampling
numbers, 376-380 , from infinite popula-
tions, 380 ; from hypothetical popula-
tions, 380-1 , purposive, 382-4
— , of attributes, 386-412 ; simple, 386-7 ;
mean and s d. m, 387-390 ; standard
error, 390 ; case where parent propor-
tion unknown, 390-4 , limitations of
simple sampling, 394-6 , applications,
396-400 ; non-simple, 400-7 ; Lexis’
approach, 407-8

— , of variables, large samples, generally,
415-458 , samphng distribution, 414-9 ;
simple sampling, 419-420 ; approxi-
mations, 420-1 ; standard error, '421 ;
of quantiles, 421-6 , of semi-inter-
quartile range, 427-8 ; of arithmetic
mean, 428 , means of two samples,
429-30 ; non-simple sampling, 430-3 ;
standard errors of moments, 437-442 ;
of variance, 442 ; of standard deviation,
442-6 ; two samples, 446 , of moments,
447-8 ; of coefficient of variation, 448-
450 ; of and ^ 2 ^ 450-1 ; of correla-
tion coefficient, 451-3 , of regression,
453 ; of correlation ratio and multiple
correlation coefficients, 453 , of coeffi-
cent of association, 454 , of coefficient
of contingency, 454 ; of Spearman’s p,
454-5 ; of Kendall’s r, 455-6
— , of variables, small samples, 482-502 ,
estimates, 482-4 ; degrees of freedom of,
484-5 ; tests of significance, 485 ;
assumption of normality, 485-7 ; t-
distribution, 487-492 ; significance of
regressions, 492-3 , Fisher's distribution
493-5 ; correlation coefficient, 495-9 ;
correlation ratio, 517-9 ; linearity of
regression, 519-520 ; multiple correla-
tion coefficent, 521-2. See also Analy-
sis of Vanance

— , practical problems, 530-554 ; size of
unit, 530-2 ; stratified .samplmg, 533 ;
sampling fractions, 533-9 ; , systemaric -
samplin^,^ 542 , quota sampling, 542 ;
sequenrial sarapflng* 543 ; examination
oT samples, 544 ; corrections for pessixn.

INDEX

699

ism, 544-5 ; duplicated enumeration,
545-7 , bias, 547-550 , the vanity effect,
550 ; the sympathy effect, 550-1 ,
methods of minimising distorted res-
ponse, 551-2

Saunders, Miss E R , data cited from, 29
Scale reading, bias in, (Table 4 4), 74
Scarlet fever, deaths from (Table 4 11,
Figure 4.11), 89, 90, mean, 108,
median, 113

Scatter diagram, 211-2, generalised, 297
Scottish Milk Records Association, 453
Screws, measurements on, (Table 4 3), 72
Seasonal effects, in time series, 624-7
Semi-mterquartile range, see Quartiles
Semi-invariants, seminvariants, see
Cumulants

Sequential sampling, 543

Serial correlation, 639

Shakespeare, W., use of word statist”,

XVI

Sheep population, (Table 26 1), 612 ;
(Figure 26.1), 613 , trend line fitted to,
620-2, (Figure 26 5), 622 ; variate-
differences of, 632-3 ; residual after
trend-elimination (Table 27.1), 640;
serial correlations (Table 27 4), 650',
correlogram (Figure 27 4), 650 ; auto-
regressive scheme for, 653-4 , residual
variance, 655

Sheppard, W F., corrections for grouping,
133-4, 158-9 ; theorem on normal
correlation, (Exercise 10 4), 252
Shipping-freights, index-number of, 603-4
Significance levels, see Levels of Signific-
ance

Silvey, R. J., on sampling for radio
audition, 550

Simple interpolation, 559-561
Simple sampling, see Samphng of Attri-
butes, Sampling of Variables
Sinclair, Sir John, use of words
” statistical ” statistics ”, xvii
Size of sampling umt, 530-2
Skew frequency-distnbutions, 83-7
Skewness, 83-7 ; measure of, 162-3 ;
standard error of Pearson’s measure of,
450

Small chances, see Poisson distribution
— , samples, see Samphng of variables,
small samples

Soil, relationship between temperature
and loss of weight, 352-7
Southey, R , (Table 4 12), 94
Spahlmger vaccme for tuberculosis in
cattle, 472

Spearman, C , theorems on correlation,
327-8 ; footrule,” footnote, 262 ; see
Rank correlation

Spencer’s formulae for graduation, 623
Spurious correlation in indices, 330
Standard deviation, see Deviation,
standard

Standard error, see Error, standard
— , error of a particular statistic, see under
that statistic or under Error, standard
Standardisation of death-rates, 335-7
” Statist,” occurrence of word in Shakes-
peare and Milton, xvi
” Statistic,” definition, footnote, 414
Statistical senes interpolation, ofi 561-3
Stature, correlation m father and son
(Table 9 3), 202, (Figure 9 3), 210,
regression lines (Figure 9 8), 218;
constants of (Exercise 9 3), 235, correla-
tion ratios, 258-9 ; test for normahty,
243-8 , test for isotropy, 249-250 ;
standard error of correlation, 452
Stature of males in the Umted Kingdom
(Table 4.7), 82 ; (Figure 4.7), 83 ;
mean, 106-7; median, 112-3, means
and medians of constituent countnes
(Exercise 5 1), 122 , standard deviation,
131-2 ; mean deviation, 139-140 ;
quartiles, 141-2 ; s, d. and m. d. of
constituent countries (Exercise 61),
148 ; third and fourth moments,
153-5, 159; ^jand/Jg, 160; skewness,
162 , kurtosis, 164 ; cumulants, 165 ;
normal curve fitted to (Figure 8.3), 189 ;
standard errors of mean, 428 ; of
median, 425-6 , of deciles, 426 ; of
standard deviation, 444 , of third and
fourth moments, 447-8
Stigmatic rays in poppies, see Poppies
Stirling, James, approximation to
factorial, 179

Sxratified samphng, 371, 382-4, 533-542
Student ” (W. S. Gosset), mnemonic for
kurtosis, 164 ; standard deviation of
Spearman’s p, 455 ; on Lanarkshire
milk experiment, 544
” Student’s ” distribution, see ^-distnbu-
tion

Sub-division of intervals, in interpolation,
571-5

Subnormal dispersion, in Lexis' sense, 408
Sugar beet, determination of sugar
content. 383-4

Sunspots, oscillations in Wolfer’s numbers
(Table 26 4), 615 , (Figure 26.4), 616 ;
as autoregressive series, 656
Supernormal dispersion, in Lexis’ sense,
408

Sur- and super-tax (Table 4.5), 77 ;

quantiles (Exercise 6.3), 149
Sympathy effect, in sampling, 550-1
Systematic sanipling, 542

t-DiSTRiBUTioN, 487-8 ; applications, to
testing a mean, 489 ; comparison of
two means, 490-2 ; regression
coefficients, 492-3 ; test of Spearman’s
p, 455 ; test of product-moment
correlations, 499

700

THEORY OF STATISTICS

Tabulation of data, 4, 59-61, 72-7, 201-3
Tangential interpolation, 579
Temperature and loss of weight in soil.
Percentage

Tests of significance, see Sampling of
variables, small samples
Tetrachoric y, 270-1 , different from
product-moment, 272 ; standard error
of, 452

Thiele, T. N , second footnote, 164
Ticket sampling, 375
Tied ranks, 264-6

Time-reversal test, in index-numbers,
590-2

Time-senes, generally, 610-661 , examples
of, 611-6; trend, 616-7; moving
averages, 617-624 , elimination of
seasonal effects, 624-7 , effect of trend-
elimination, 627-631 , variate differenc-
ing, 631-3 , tests for randomness, 638-9,
serial correlation, 639-641 , penodogram
analysis, 641-5 , autoregressive series,
645-651 , correlogram, 651-4 , proper-
ties of autoregressive series, 654-6 ,
period of an oscillation, 656-8
Tippett, LHC, sampling numbers, 376
Tocher, J. F,, data cited from, (Table 9 4),
204 ; correlation of milk-yield and
butter fat, 452

Trend, 616 ; determination by moving
averages, 617-624 , effect of elimination
on harmonic component, 629 ; on
random component, 630 ; variate-
differences, 631-3
Trough, in time-series, .638
Truncated frequency-distribution, 91-2
Tschuprow, A. A., coefficient of contin-
gency, 54-6

Tuberculosis m cattle, vaccine for, 472
Turning-point, in time-senes, 638
Type, of array, 199

Ultimate classes, 5-6, 7-8
Undertakings, Electricity, see Electricity
Unit, size of, in correlation, 310-3 , in
sampling, 531-2

U-shaped distributions, 90-1, 93

Value of estates, (Table 4.12), 94 ,
(Figure 4.13), 92
Vanity effect, in sampling, 550
Variables, theory of, generally, 69ff ,
sampling of, see Sampling of variables
Variance, definition, 127 ; standard error
of, 442 ; estimates of, 483 ; Analysis of,
see- Analysis

Variate, definition, footnote, 69
Variate-difference method, 317-9 ; in
determining moving averages, 631-3

Variation, coefficient of, 143-4 ; standard
error of, 448-450

Velocity-distance relation m nebulae,
(Table 15 1), 340, (Figure 15.1), 341,
347-8

Volume of exports, index-number of, 606

Wages, of labourers, see Agricultural
Labourers, Earnings
Wald, A , sequential =rm-]-r:r. 543
Weather and crops, corr i »i , 320-1
Weight of criminals, (Table 3 6), 64
Weight of males m the United Kingdom,
(Exercise 4 6), 100 ; mean, median and
mode (Exercise 5 3), 122, sd, md.,
quartiles (Exercise 6 2), 149 , moments,
skewness (Exercises 7 1
and 7.2), 167 ; standard error of mean
(Exercise 18 5) 435 , of median and

quartiles (Exercise 18 4) 434 , of

standard deviation (Exercise 19 1), 457
Weldon, W F R , Dice
Wheat, yields of (Table 13 1) 311, 493, 494 ;
prices (Table 25.1), 591 , Beveridge
price-index, 645, (Exercise 27 8), 660
— shoots, distribution of (Table 16 1),
372

Whitaker, Lucy, data cited from (Exercise
8.17), 198

Whiting, Madeleine H , data cited from
(Table 3 6), 64

Wholesale prices, index-number of, 598-9
Willis, J. C , data regarding ChrysomehdcB,
(Table 4 13), 95

Wireless licences, see Mental defectives
Wolfer’s sunspot numbers, (Table 26 4 1 ,
615

Woo, T. L , data cited from (Exercise 3 10),
67 8

j Yates, F , data cited from (Table 16.1)

! 372 ; on farm survey, 536-8 , Sampling

' methods, footnote, 542
I Yields, of gram, (Exercise 4,5 {c) ), 99 ;
(Table 22.5), 509 ; of potatoes,

(Exercise 27.1), 659 ; of nulk, see milk-
yields

Yule, G. Udny, passim , data cited from,
cholera, 25-6, 27-8 ; poppies (Table
4 3), 71 ; reading a scale (Table 4 4),
74 ; (Table 4.13), 95 ; duckweed (Table
9.6), 226 ; experiments on 476 ;
judgment of tint (Exercise 20 5), 479 ;
correlation, 520 ; sunspot numbers
(Table 26 4), 615

INDEX

701

2-DiSTRiBUTioN see Fisher’s distribution
Zimmerman, E. A, W , use of words
“ statistics/' " statistical " in English,
xvi-xvu

/^-coefficients, 159

B-function, use of , m ^ test, 494

y coefficients, 159

p, see Rank Correlation

T, see Rank Correlation

(chi-square, chi-squared) generally,
459-481 , constraints in, 461 ; degrees
of freedom 461-2 , definition, 463 ; test
when theoretical frequencies known a
pnon, 465-8 ; properties of distribution,
468-9 , conditions for applicability of
test, 469-470 , additive property,
473-4 , test when data are used to
estimate theoretical frequencies, 474-6 ,
experiments on distnbution 476-7 ,*
goodness of fit, 477

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